1. Introduction

Let \(P_m\), \(C_m\), \(K_m\), \(K_m-I\), \(K_{m,m}-I\) denote path of length \(m\), cycle of length \(m\), complete graph on \(m\) vertices, complete graph on \(m\) vertices minus a 1-factor and complete bipartite graph on \(2m\) vertices minus a 1-factor respectively. All graphs considered in this paper are simple, finite and connected. We refer to the book [1] for graph theoretic terminology used in this article. A graph is said to be it decomposed into subgraphs \(H_1\) and \(H_2\) which is denoted by \(G= H_1 \oplus H_2\), if \(G\) is the edge disjoint union of \(H_1\) and \(H_2\). Assume that \(G= H_1 \oplus H_2 \oplus \cdots \oplus H_k\) and if each \(H_i\), \(1 \leq i \leq k\), is a path or cycle in \(G\), then the collection of edge-disjoint subgraphs of \(G\) denoted by \(\psi\) is called a path decomposition of \(G\). If each \(H_i\) is a path in \(G\) then \(\psi\) is called an acyclic path decomposition of \(G\). The minimum cardinality of a path decomposition of \(G\), denoted by \(\pi(G)\), is called the path decomposition number and the minimum cardinality of an acyclic path decomposition of \(G\), denoted by \(\pi_a(G)\), is called the acyclic path decomposition number of \(G\). If \(P=(x_1,x_2,...,x_m)\) is a path in a graph \(G\), then the vertices \(x_2,x_3,...,x_{m-1}\) are called the internal vertices of \(P\) and \(x_1\), \(x_m\) are called external vertices of \(P\). Here, by a first vertex and end vertex of path \(P\) we mean the vertices \(x_1\) and \(x_m\) respectively. Let \(P=(x_1,x_2,...,x_m)\) and \(Q=(y_1,y_2,...,y_m)\) be two paths in \(G\), by joining \(x_1\) to \(y_1\) (\(x_m\) to \(y_m\), respectively) we obtain the path \(R=(y_m,y_{m-1},...,y_1,x_1,x_2,...,x_m)\) \(\big (R=(x_1,x_2,...,x_m,y_m,y_{m-1},...,y_1)\), respectively \(\big )\).

Definition 1.1. The Cartesian product \(G \ \Box \ H\) of two graphs \(G\) and \(H\) is a graph with vertex set \(V(G)\times V(H)\) in which \((x_1,y_1)\) and \((x_2,y_2)\) are adjacent if one of the following condition holds:

• \(x_1= x_2\) and \(\{y_1,y_2\}\in E(H)\).
• \(y_1= y_2\) and \(\{x_1,x_2\}\in E(G)\).

The graphs \(G\) and \(H\) are known as the factors of \(G \ \Box \ H\).

Suppose we are dealing with \(m\)-copies of a graph \(G\) we denote these \(m\)-copies of \(G\) by \(G^i\), where \((i=1,2,3,...,m)\).
The Cartesian product graph \(G \ \Box \ H\) may also be viewed as the graph obtained from \(G\) by replacing each vertex \(i \in V(G)\) by a copy \(H^i\) (say) of \(H\) and each of its edges \(\{i, k\}\) with \(|V (H)|\) edges joining corresponding vertices of \(H^i\) and \(H^k\).
Henceforth, for any vertex \(i \in V(G)\) we refer the copy of \(H\), denoted by \(H^i\), in \(G \ \Box \ H\) corresponding to the vertex \(i\) as the \(i^{th}\) copy of \(H\) in \(G \ \Box \ H\).
The problem of finding \(C_k\)-decomposition of \(K_{2n+1}\) or \(K_{2n}-I\) where \(I\) is a 1-factor of \(K_{2n}\), is completely settled by Alspach, Gavlas and Sajna in two different papers (see [2, 3]). Obviously, every graph admits a decomposition in which each subgraph \(H_i\) is either a path or a cycle. Gallai conjectured that the minimum number of paths into which every connected graph on \(n\) vertices can be decomposed into is not less than \(\lceil \frac{n}{2} \rceil\) (see [4]). A significant contribution to the parameter \(\pi\) was by Lovasz [4] when he proved that a graph on \(n\) vertices can be decomposed into \(\lfloor \frac{n}{2} \rfloor\) paths and cycles. Harary introduced the parameter \(\pi_a\), this was further studied by Harary and Schwenk in [5] when they considered the evolution number of the path number of a given graph. Staton it et al. in [6, 7] provided further results on path numbers and considered the case of the tripartite graphs. P\'eroche [8] gave some results on the path numbers of certain multipartite graphs. Arumugam and Suseela [9] shed some lights on the acyclic path decomposition of unicyclic graphs. A recent work by Arumugam it et al. [10] studied the parameter \(\pi\) and further determined the value of \(\pi\) for some graphs. They also provided some bounds for \(\pi\) and characterize graphs attaining the bounds. Furthermore, they proved that the difference between the parameter \(\pi\) and \(\pi_a\) can be arbitrary large.
In this paper, we determine the value of \(\pi\) for the graph \(K_n-I\), \(K_{n,n}-I\) and the Cartesian products \(P_m \ \Box \ C_n\) and \(C_m \ \Box \ C_n\). In addition, we classify the graphs that attain some of the bounds mentioned in [10].

2. Path decomposition number of \(K_n-I\) and \(K_{n,n}-I\)

Theorem 2.1 [2] For even integers \(m\) and \(n\) with \(4 \leq m \leq n\), the graph \(K_n-I\) decomposes cycles of length \(m\) if and only if the number of edges in \(K_n-I\) is a multiple of \(m\)

Lemma 2.2. [11] Let \(m \equiv 2(mod \ 4)\), \(n \equiv 1(mod \ 2)\) and \(6 \leq m \leq 2n\). Then \(C_m |K_{n,n}-I\) if and only if \(m|n(n-1)\).

Theorem 2.3. Given the graph \(K_n-I\), where \(n\) is even, the minimum path decomposition number for \(K_n-I\) is \(\frac{n-2}{2}\).

Proof. The graph \(K_n-I\) has \(n\) vertices and \(\frac{n(n-2)}{2}\) edges. The largest cycle which is a subgraph of \(K_n-I\) is a cycle of order \(n\). Now, by Theorem 2.1, \(C_n |K_n-I\). We only need to know the number of copies \(C_n\) that can be gotten from \(K_n-I\), which is \(\frac{n-2}{2}\). Thus, we have \(\frac{n-2}{2}\) copies of \(C_n\) in \(K_n-I\). Therefore, \(\pi (K_n-I)=\frac{n-2}{2}\).

Lemma 2.4. If \(n \geq 4\) and an even integer, then \(K_{n,n}-I\) is \(\Big ( \frac{n-2}{2} C_{2n},nP_2 \Big )\)-decomposable.

Proof. Let \(X=\{1^1,2^1,3^1,...,n^1\}\) and \(Y=\{1^2,2^2,3^2,...,n^2\}\) form the column set of vertices in \(K_{n,n}-I\). Also, two vertices \(a^i\) and \(b^j\), has an edge in \(K_{n,n}-I\), if \(a \neq b\) and \(i \neq j\), \(i< j=2\). Since \(n\) is even, the degree of each vertex in \(K_{n,n}-I\) is odd.
Next, remove the edges $$ E(a^i, b^j)= \left\{\begin{array}{cc} (a,n-a+1) ,& \; a=1,2 \\ (a,a-2), & \; a=3,4,5,...,n \end{array}\right. ,a^i \in X, b^j \in Y $$ which are exactly \(n\) number of \(P_2\)'s. By removal of these edges, each vertex in \(K_{n,n}-I\) would be of even degree. In total, we have \(n(n-2)\) edges. At this point, we need to show that the subgraph \((K_{n,n}-I) \setminus E(a^i,b^j)\) admits a \(C_{2n}\) decomposition.
Now, by \(C_{2n}^r\), \(r \leq 1\), we mean the \(r^{th}\) copy of \(C_{2n}\) in \((K_{n,n}-I) \setminus E(a^i,b^j)\). With exception of \(C_{2n}^1\), all other \(C_{2n}^r\), \(r>1\), follow a similar pattern. The construction of these cycles of order \(2n\) is given below. \begin{align*} C_{2n}^1 & =1^1,2^2,3^1,4^2,...,(n-1)^1,n^2,(n-2)^1,(n-3)^2,(n-4)^1, (n-5)^2,\nonumber\\ & \ \ \ \ 2^1,1^2,n^1,(n-1)^2,1^1. \end{align*} For \(r=2,3,4,...,\frac{n-2}{2}\) we have that \begin{align*} C_{2n}^r & =1^1,(2r-1)^2,n^1,(2r-2)^2,(n-1)^1, (2r-3)^2,(n-2)^1,...,1^2, \nonumber\\ & \ \ \ \ (n-2r+2)^1,(n-1)^2,(n-2r+1)^1,n^2,(n-2r)^1,(n-2)^2, \nonumber\\ & \ \ \ \ (n-2r-1)^1,(n-3)^2,(n-2r-2)^1,(n-4)^2,...,(2r)^2,1^1. \end{align*} From the above construction, we conclude that the graph \((K_{n,n}-I) \setminus E(a^i,b^j)\) admits a \(C_{2n}\) decomposition. Clearly, \(r=\frac{n-2}{2}\) and thus \(C_{2n}|\big\{K_{n,n} -I \setminus E(a^i,b^j)\big \}=(C_{2n} \oplus C_{2n} \oplus C_{2n} \oplus \cdots \oplus \frac{n-2}{2} C_{2n} )\). Finally, we have that \(K_{n,n}-I\) is \(\Big ( \frac{n-2}{2} C_{2n},nP_2 \Big )\)-decomposable. Hence the proof.

Theorem 2.5. For the complete bipartite graph \(K_{n,n}-I\), we have that $$ \pi (K_{n,n}-I)= \left\{\begin{array}{cc} \frac{n-1}{2} ,& \; if \ n \ is \ odd \\ \frac{3n-2}{2} ,& \; otherwise \end{array}\right. $$

Proof. The graph \(K_{n,n}-I\) has \(2n\) vertices and \(n(n-1)\) edges. The largest cycle which is a subgraph of \(K_{n,n}-I\) is a cycle of order \(2n\). We now prove this theorem in two cases.

Case 1: when \(n\) is odd
By Lemma 2.2, \(C_{2n} |K_{n,n}-I\). We only need to know the number of copies of \(C_{2n}\) that can be obtained from \(K_{n,n}-I\), which is \(\frac{n-1}{2}\). Therefore, \(\pi (K_{n,n}-I)=\frac{n-1}{2}\).

Case 2: when \(n\) is even
By Lemma 2.4, the graph \(K_{n,n}-I\) can be decomposed into \(\frac{n-2}{2}\) copies of \(C_{2n}\) and \(n\) copies of \(P_2\). Since no vertex is repeated in these \(n\) copies of \(P_2\), we have that \(\pi (K_{n,n}-I)=\frac{3n-2}{2}\). The proof of this theorem is complete.

To end this section we now give the following remark. This remark is immediate from Theorem 2.3 and Theorem 2.5.

Remark 2.6. In [10], it was mentioned that every graph \(G\) which is Hamiltonian cycle decomposable attains the bound that \(\pi (G) \geq \lceil \frac{\Delta}{2} \rceil\). This is true as we see from Theorem 2.3 and Theorem 2.5 that the complete graph minus a one-factor and the complete bipartite graph \(K_{n,n}-I\), where \(n\) is odd, attains this bound. Now, when \(n\) is even in \(K_{n,n}-I\) we have \(\pi (K_{n,n}-I)=\frac{3\Delta+1}{2}\).

3. Path decomposition number of \(P_m \ \Box \ C_n\) and \(C_m \ \Box \ C_n\)

Theorem 3.1. Let \(m\) and \(n\) be positive integers then $$\pi(P_m \ \Box \ C_n )= \pi_a (P_m \ \Box \ C_n )=n$$.

Proof. First we give the construction of \(P_{mn}\) paths by constructing Hamilton paths of order \(n\) in each copy of \(C_n\) in \(P_m \ \Box \ C_n\). Let \(i\) be an odd number, in each copy of \(C_n^i\), join the end vertex of the Hamilton path in the \(i^{th}\) copy with the end vertex of the \(C_n^{i+1}\) copy of \(P_m \ \Box \ C_n\). Similarly, suppose \(i\) is even, in each copy of \(C_n^i\), join the first vertex of the Hamilton path in the \(i^{th}\) copy with the first vertex of the \(C_n^{i+1}\) copy of \(P_m \ \Box \ C_n\).

Next, for each internal vertex in the Hamilton path, join the vertices \(x_j^i\) and \(x_j^{i+1}\), \(1 \leq i \leq m\), \(i\) is calculated in modulo \(m\) and \(2 \leq j \leq n-1\). By this, we have \(n-2\) copies of \(P_m\) in \(P_m \ \Box \ C_n\).
Lastly, the left out edges which has not been covered by the path \(P_{mn}\) and the \(n-2\) copies of \(P_m\) form a path of order \(2m\). So we have that \(\pi(P_m \ \Box \ C_n)=\pi_a (P_m \ \Box \ C_n)=n\).

Remark 3.2. Since the Cartesian product of graph is commutative, the result in Theorem 3.1 holds for the graph \(C_m \ \Box \ P_n\) where \(m\) and \(n\) are positive integers.

Theorem 3.3. Let \(m\) and \(n\) be positive integers such that \(3 \leq n \leq m\), then \(\pi (C_m \ \Box \ C_n)=n\).

Proof. Since both \(m\) and \(n\) are positive integers, the proof of this theorem is split in two cases.

Case 1: when \(m\) is even and \(n \geq 3\).
First we give the construction of \(C_{mn}\) cycles by constructing Hamilton paths of order \(n\) in each copy of \(C_n\) in \(C_m \ \Box \ C_n\). Let \(i\) be an odd number, in each copy of \(C_n^i\), join the end vertex of the Hamilton path in the \(i^{th}\) copy with the end vertex of the \(C_n^{i+1}\) copy of \(C_m \ \Box \ C_n\). Similarly, suppose \(i\) is even, in each copy of \(C_n^i\), join the first vertex of the Hamilton path in the \(i^{th}\) copy with the first vertex of the \(C_n^{i+1}\) copy of \(C_m \ \Box \ C_n\).
Next, for each internal vertex in the Hamilton path, join the vertices \(x_j^i\) and \(x_j^{i+1}\), \(1 \leq i \leq m\), \(i\) is calculated in modulo \(m\) and \(2 \leq j \leq n-1\). By this, we have \(n-2\) copies of \(C_m \ \Box \ C_n\).
Now, notice that the left out edges which has not been covered by the cycle \(C_{mn}\) and the \(n-2\) copies of \(C_m\) form a cycle of order \(2m\). So we have that \(\pi (C_m \ \Box \ C_n)=n\).

Case 2: when \(m\) is odd and \(n \geq 3\).
Here, we first give the construction of \(C_{mn-1}\) cycles. For \(1 \leq i \leq m-2\), construct Hamilton paths of order \(n\) in each \(C_n^i\) copy in \(C_m \ \Box \ C_n\). Suppose \(i\) is odd, in each copy of \(C_n^i\) join the end vertex of the Hamilton path in the \(i^{th}\) copy with the end vertex of the \(C_n^{i+1}\) copy of \(C_m \ \Box \ C_n\). In the same way, if \(i\) is even, in each copy of \(C_n^i\) join the first vertex of the Hamilton path in the \(i^{th}\) copy of \(C_n^i\) with the first vertex of the \(C_n^{i+1}\) copy of \(C_m \ \Box \ C_n\). This gives a path of order \(n(m-2)\).
Next, let \(x\) be the first vertex in the \(C_n^{m-1}\) copy of \(C_m \ \Box \ C_n\). Now, construct a path \(P_{n-1}\) from \(C_n^{m-1} \setminus x\). Join the end vertex of \(C_n^{m-1}\) copy to the end vertex of \(C_n^{m-2}\) copy of \(C_m \ \Box \ C_n\). Since \(x\) is removed from \(C_n^{m-1}\), join the second vertex \(x_2^{m-1}\) of \(C_n^{m-1}\) to the second vertex \(x_2^m\) of \(C_n^m\) and then move in a clockwise direction to the first vertex in the \(m^{th}\) copy of \(C_m \ \Box \ C_n\). To get the desired \(C_{mn-1}\) cycle, join the first vertex \(x_1^m\) of \(C_n^m\) to \(x_1^1\) of \(C_n^1\) in the graph \(C_m \ \Box \ C_n\).
Furthermore, aside the second vertex, each internal vertex \(x_j^i\) and \(x_j^{i+1}\), \(1 \leq i \leq m\), \(i\) is calculated in modulo \(m\) and \(3 \leq j \leq n-1\) when joined in all other copies of \(C_n\) results to \(n-3\) copies of \(C_m\) in \(C_m \ \Box \ C_n\).
The left out edges which have not been covered by the cycle \(C_{mn-1}\) and the \(n-3\) copies of \(C_m\) form cycles \(C_{m+2}\) and \(C_{2m-1}\). We now give the construction of cycles \(C_{m+2}\) and \(C_{2m-1}\) as follows. By \(x_j^i\) we mean the \(j^{th}\) vertex of \(C_n\) in copy \(i\) of the graph \(C_m \ \Box \ C_n\). \begin{align*} C_{m+2}=x_2^1,x_2^2,x_2^3,...,x_2^{m-1},x_1^{m-1},x_1^m,x_2^m,x_2^1.\nonumber\\ C_{2m-1}=x_n^1,x_1^1,x_1^2,x_n^2,x_n^3,x_1^3,...,x_1^{m-1},x_n^{m-1},x_n^m,x_n^1.\nonumber \end{align*} Therefore we have that \(\pi (C_m \ \Box \ C_n)=n\). This completes the proof.

We now conclude this section with the following remark.

Remark 3.2. We note here in this section that although Arumugam et al. in [10] gave a relationship between the path decomposition number (or acyclic path decomposition number, as the case maybe) and the maximum degree \(\Delta\) of some graphs, we note that for the product \(G \ \Box \ H\) there is no such relationship since the parameters \(\pi (G \ \Box \ H)\) and \(\pi_a (G \ \Box \ H)\) do not depend on \(\Delta(G \ \Box \ H)\).

4. Conclusion and future work

So far in this work we have provided the path decomposition number for \(K_n-I\), \(K_{n,n}-I\) and the product \(P_m \ \Box \ C_n\) and \(C_m \ \Box \ C_n\). The question for determining the acyclic path decomposition number for these graphs certainly deserves attention. As a future work, we intend to provide the acyclic path decomposition number for these graphs and possibly look into other types of product graphs, e.g. lexicographic and tensor products.

Competing Interests

The authors declare that they have no competing interests.

1. Introduction

The nowhere-zero flows were defined by Tutte in [1]. The definition of nowhere-zero flows on signed graphs naturally comes from the study of embedding of graphs in non-orientable surfaces, where nowhere-zero flows emerge as the dual notion to local tensions. There is a close relationship between nowhere-zero flows and circuit covers of graphs as every nowhere-zero flow on a graph \(G\) determines a covering of \(G\) by circuits. This relationship is maintained for signed graphs, although a signed version of the definition of circuit is required.

Let \(G\) be a directed graph. A nowhere-zero flow on \(G\) is an assignment of non-zero integers to each edge of \(G\) such that for every vertex the Kirchhoff current law holds, which says that, the sum of the values of incoming edges is equal to the sum of the values of outgoing edges. A nowhere-zero \(k\)-flow is a nowhere-zero flow using edge labels with maximum absolute value \(k-1\). Since for a directed graph that admits nowhere-zero flows is independent of the choice of the orientation, therefore one may consider such concept over the underlying undirected graph. A celebrated conjecture of Tutte in 1954 says that every bridge less graph has a nowhere-zero 5-flow. Jaeger showed in 1979 that every bridgeless graph has a nowhere-zero 8-flow [2], and Seymour proved that every bridgeless graph has a nowhere-zero-6-flow [3]in 1981. However the original Tutte conjecture remains open, see for more result [4,5, 6].

As an analogous concept of a nowhere-zero flow for directed graphs, we consider zero-sum flow number for undirected graphs in this paper.

Definition 1.1. For an undirected graph \(G\), a zero-sum flow is an assignment of non-zero integers to the edges such that the sum of the values of all edges incident with each vertex is zero. A zero-sum \(k\)-flow is a zero-sum flow whose values are integers with absolute value less than \(k\).

Note that from algebraic point of view finding such zero-sum flows is the same as finding nowhere zero vectors in the null space of the incidence matrix of the graph. Akbari et al. raised a conjecture for zero-sum flows similar to the Tutte $5$-flow Conjecture for nowhere-zero flows as follows:

Conjecture 1. (Zero-Sum 6-Flow Conjecture) If \(G\) is a graph with a zero sum flow, then \(G\) admits a zero-sum \(6\)-flow.

It was proved in 2010 by Akbari et al. in [7] that the above zero-sum 6-flow Conjecture is equivalent to the Bouchet 6-flow Conjecture for bidirected graphs. In literatures a more general concept flow number, which is defined as the least integer \(k\) for which a graph may admit a \(k\)-flow, has been studied for both directed graphs and bi directed graphs. Wang and Hu in [8, 9] extend the concept in 2011 to the undirected graphs and call it zero-sum flow number, and also considered general constant-sum flows for regular graphs.

In the study of nowhere-zero flows of directed graphs (bidirected graphs) one considers a more general concept, namely, the least number of \(k\) for which a graph may admit a \(k\)-flow. In [9], Wang and Hu considered similar concepts for zero-sum \(k\)-flows. In the study of nowhere-zero flows of directed graphs(bidirected graphs) one considers a more general concept, namely, the least number of \(k\) for which a graph may admit a \(k\)-flow. In [9] Wang and Hu consider similar concepts for zero-sum \(k\)-flows.

Definition 1.2. Let \(G\) be an undirected graph. The zero-sum flow number \(F(G)\) is defined as the least number of \(k\) for which \(G\) may admit a zero-sum \(k\)-flow. \(F(G) = \infty\) if no such \(k\) exists.

It is well known that grids are extremely useful in all areas of computer science. One of the main usage, for example, is as the discrete approximation to a continuous domain or surface. Numerous algorithms in computer graphics, numerical analysis, computational geometry, robotics and other fields are based on grid computations. In [10] and [11] the authors calculated the zero-sum flow number of triangular and hexagonal grids. In this paper, we calculate zero-sum flow number of Octagonal grid, generalized prism and book graph.

2. Zero-Sum Flow Number of Octagonal Grid

In [12], Kamran et al. considered this octagonal grid and computed the exact value of total edge irregularity strength for octagonal grid. For \(n,m \geq2\) we denote octagonal grid by \(O_n^m\), the planar map labeled as in Figure 1 with \(m\) rows and \(n\) columns of octagons. The symbols \(V (O_n^m)\) and \(E(O_n^m)\) denote the vertex set and the edge set of \(O_n^m\) , respectively

\(V (O_n^m ) = \{x_i^j ; 1 \leq i \leq 2n-1,\) \(i\) odd and \(1\leq j \leq 3m + 1\} \cup\{ x_i^{3j-2}\) \(1 \leq i \leq 2n\); \(i\) even and \(1 \leq j \leq m + 1\}\cup \{ x_{2n}^{3j-1},\ x_{2n}^{3j}; 1 \leq j \leq m\}\) \begin{eqnarray*} V (O_n^m ) &=& \{x_i^j ; 1 \leq i \leq 2n-1, \;i\;\textrm{odd and}\; 1\leq j \leq 3m + 1\} \\ &\cup&\{ x_i^{3j-2} ; \;1 \leq i \leq 2n; \;i \;\textrm{even and}\;1 \leq j \leq m + 1\}\\ &\cup& \{ x_{2n}^{3j-1},\ x_{2n}^{3j}; 1 \leq j \leq m\}\\ E(O_n^m ) &=& \{x_i^j x_i^{j+1} ; 1 \leq i \leq 2n-1; \;i\; \textrm{odd and} \;1 \leq j \leq 3m\} \\ &\cup & \{x_i^{3j-2}x_{i+1}^{3j-2}; \; 1 \leq i \leq 2n-1; \;i\; \textrm{odd and} \;1 \leq j \leq m + 1\} \\ &\cup & \{x_i^{3j-2} x_{i+1}^{3j-1}; 1 \leq i \leq 2n-2; \;i\; \textrm{even and} \;1 \leq j \leq m\} \\ &\cup & \{ x_i^{3j}x_{i-1}^{3j+1}; 3 \leq i \leq 2n -1; \;i\;\textrm{odd and} \; 1 \leq j \leq m\} \\ &\cup & \{ x_{2n}^j x_{2n}^{j+1}; 1 \leq j \leq 3m\} \end{eqnarray*} $$|V(O_{n}^{m})|=(4m+2)n+2m\,\, \ and\, \,\ |E(O_{n}^{m})|=(6m+1)n+m$$

Theorem 2.1. The zero-sum flow number \(F(O_n^m)\) of \(O_n^m\) is \(3\) for all \(n,m\geq2\).

Proof. Note that there are \(4n+4m\) vertices of degree 2 and \(4mn-2n-2m\) vertices of degree \(3\) in \(O_n^m\), so a zero-sum flow edge assignment from \(\{-1,1\}\) is not possible. Therefore \(F(O_n^m)\) is at least \(3\). To prove the converse inequality we will consider the following edge labeling \(\varphi:E(O_n^m)\rightarrow\{1,-1,2\}\). $$ \varphi(x_i^{3j-2}x_{i+1}^{3j-2})=\left\{ \begin{array}{ll} 1, & \hbox{\(j=1,\ m+1\), \(i\) is odd and \(1\leq i\leq2n-1\)} \\ 2, & \hbox{\(2\leq j\leq m\), \(i\) is odd and \(1\leq i\leq2n-1\)} \end{array} \right.$$ $$ \varphi(x_i^{3j-1}x_{i}^{3j})=\left\{ \begin{array}{ll} 1, & \hbox{\(i=1,\ 2n\), \(1\leq j\leq m\)} \\ 2, & \hbox{\(1\leq j\leq m\), \(i\) is odd and \(3\leq i\leq2n-1\)} \end{array} \right.$$ For \(i\) odd, \(1\leq i\leq2n-1\) and \(1\leq j\leq m\), $$\varphi(x_i^{3j}x_{i}^{3j+1})=\varphi(x_i^{3j-2}x_{i}^{3j-1})= -1,$$ For \(i\) even, \(1\leq i\leq2n\) and \(1\leq j\leq m\), $$\varphi(x_i^{3j-2}x_{i+1}^{3j-1})= -1,$$ For \(i\) odd, \(3\leq i\leq2n\) and \(1\leq j\leq m\), $$\varphi(x_i^{3j}x_{i-1}^{3j+1})= -1,$$ We can see that \(\varphi\) is an edge labeling from \(E(O_n^m)\) to \(\{1,-1,2\}\). Now we will find the weight of each vertex and the weight of a vertex is the sum of all labels of edges adjacent to it. \begin{eqnarray*} wt(x_i^{3j-2}) &=& \varphi(x_i^{3j-2}x_{i+1}^{3j-2})+\varphi(x_i^{3j-2}x_{i}^{3j-1}) \\ &=& 0, \quad\quad \textrm{for}\;\; 1\leq i\leq 2n-1,\; \textrm{and} \;i \;\textrm{odd}, j=1\\ wt(x_i^{3j-2}) &=& \varphi(x_i^{3j-2}x_{i+1}^{3j-2})+\varphi(x_i^{3j-2}x_{i+1}^{3j-1}) \\ &=& 0, \quad\quad \textrm{for} \;1\leq i\leq 2n,\; \textrm{and} \;i\; \textrm{even}, \;j=1 \\ wt(x_i^{3j-1}) &=& \varphi(x_i^{3j-1}x_{i}^{3j})+\varphi(x_i^{3j-2}x_{i+1}^{3j-1}) \\ &=& 0, \quad \quad \textrm{for} \;\;i=1,2n, \;\;1\leq j\leq m\\ wt(x_i^{3j}) &=& \varphi(x_i^{3j-1}x_{i}^{3j})+\varphi(x_i^{3j}x_{i}^{3j+1})\\ &=& 0, \quad \quad \textrm{for} \;i=1,2n,\; 1\leq j\leq m \end{eqnarray*} \begin{eqnarray*} wt(x_i^{3m+1}) &=& \varphi(x_i^{3m}x_{i}^{3m+1})+\varphi(x_i^{3m+1}x_{i+1}^{3m+1}) \\ &=& 0, \quad \textrm{for}\; 1\leq i\leq 2n-1,\; \textrm{and} \;i\; \textrm{odd}.\\ wt(x_i^{3m+1}) &=& \varphi(x_{i-1}^{3m+1}x_{i}^{3m+1})+\varphi(x_i^{3m+1}x_{i+1}^{3m}) \\ &=& 0, \quad \quad \;\textrm{for} \;1\leq i\leq 2n,\; \textrm{and} \;i\; \textrm{even} \\ wt(x_i^{3j+1}) &=& \varphi(x_{i-1}^{3j+1}x_{i}^{3j+1})+\varphi(x_i^{3j+1}x_{i+1}^{3j+2})+\varphi(x_i^{3j+1}x_{i+1}^{3j}) \\ &=& 0, \quad \quad \;\textrm{for} \;1\leq i\leq 2n,\; \textrm{and} \;i\; \textrm{even}, \;1\leq j\leq m-1\\ wt(x_{2i+1}^{3j-1}) &=& \varphi(x_{i+1}^{3j-2}x_{i+2}^{3j-1})+\varphi(x_{2i+1}^{3j-2}x_{2i+1}^{3j-1})+\varphi(x_{2i+1}^{3j-1}x_{2i+1}^{3j}) \\ &=& 0 , \quad\quad \textrm{for} \;1\leq i\leq n-1, \;1\leq j\leq m\\ wt(x_{2i+1}^{3j}) &=& \varphi(x_{2i+1}^{3j-1}x_{2i+1}^{3j})+\varphi(x_{2i}^{3j+1}x_{2i+1}^{3j})+\varphi(x_{2i+1}^{3j}x_{2i+1}^{3j+1}) \\ &=& 0, \quad \quad \textrm{for}\; 1\leq i\leq n-1\;, \;1\leq j\leq m\\ wt(x_{2i-1}^{3j+1}) &=& \varphi(x_{2i-1}^{3j+1}x_{2i-1}^{3j+2})+\varphi(x_{2i-1}^{3j+1}x_{2i-1}^{3j})+\varphi(x_{2i-1}^{3j+1}x_{2i}^{3j+1}) \\ &=& 0, \quad \quad \textrm{for} \;1\leq i\leq n,\; 1\leq j\leq m-1 \end{eqnarray*} These computations shows that that \(\varphi\) is indeed a zero-sum 3-flow and we get \(F(O_n^m)\leq3\). This concludes the result.

3. Zero-Sum Flow Number of Generalized Prism

The cartesian product \(G \times H\) of graphs \(G\) and \(H\) is a graph such that the vertex set of \(G\times H\) is the cartesian product \(V (G)\times V (H)\) and any two vertices \((u,u')\) and \((v,v')\) are adjacent in \(G \times H\) if and only if either \(u = v\) and \(u'\) is adjacent to \(v'\) in \(H\) , or \(u' = v'\) and \(u\) is adjacent to \(v\) in \(G\).

The generalized prism \(P_n^m\) can be defined as the cartesian product \(C_n\times P_m\) of a cycle on \(n\) vertices with a path on \(m\) vertices. If we consider a cycle \(C_n\) with \(V(C_n)=\{x_i:\ 1\leq i\leq n\}\), \(E(C_n)=\{x_ix_{i+1}:\ 1\leq i\leq n-1\}\cup\{x_nx_1\}\) and a path \(P_m\) with \(V(P_m)=\{y_j:\ 1\leq j\leq m\}\), \(E(P_m)=\{y_jy_{j+1}:\ 1\leq j\leq m-1\}\), then \(V(P_n^m)=V(C_n\times P_m)=\{(x_i,y_j):\ 1\leq i\leq n,\ 1\leq j\leq m\}\) is the vertex set of the graph \(P_n^m\) and

\begin{eqnarray*} % \nonumber to remove numbering (before each equation) E(P_n^m) &=&E(C_n\times P_m) =\{(x_i,y_j)(x_{i+1},y_j):\ 1\leq i\leq n-1,\ 1\leq j\leq m\}\\ & \cup & \{(x_n,y_j)(x_1,y_j):\ 1\leq j\leq m\}\\ & \cup &\{(x_i,y_j)(x_i,y_{j+1}):\ 1\leq i\leq n,\ 1\leq j\leq m-1\} \end{eqnarray*} is the edge set of \(P_n^m\) . So, \(|V(P_n^m)|=nm\) and \(|E(P_n^m)|=n(2m-1)\).

The generalized prism \(P_n^m\) has been studied extensively in recent years. Kuo et al. in [13] and Chiang et al. in [14] studied distance-two labelings of \(P_n^m\). Deming et al. in [15] gave complete characterization of the cartesian product of cycles and paths for their incidence chromatic numbers. Gravier et al. in [16] showed the link between the existence of perfect Lee codes and minimum dominating sets of \(P_n^m\). Lai et al. in [17] determined the edge addition number for the cartesian product of a cycle with a path. Chang et al. in [18] established upper bounds and lower bounds for global defensive alliance number of \(P_n^m\) and showed that the bounds are sharp for certain \(n,m\). In [19], Bača, et al. compute the exact value of total edge irregularity strength for generalized prism \(P_n^m\).

In following theorem we determine the exact zero-sum flow number \(F(P_n^m)\) of \(P_n^m\).

Theorem 3.1. The zero-sum flow number \(F(P_n^m)\) of \(P_n^m\) is \(3\) for all \(n\geq3\) and \(m\geq2\).

Proof. Since there are \(mn-2n\) vertices of degree \(4\) and \(2n\) vertices of degree 3 in \(P_n^m\) so \(\{-1,1\}\) assignment for the edges is not possible for the zero sum flow therefore \(F(P_n^m)\geq3\). Now we will show that \(F(P_n^m)\leq3\) and for this purpose we shall consider the following labeling \(\varphi:E(P_n^m)\rightarrow \{-1,-2,2\}\) on the edges of \(P_n^m\) graph. $$\varphi((x_i,y_j)(x_{i+1},y_j))= \left\{ \begin{array}{ll} -1, & \hbox{\(1\leq i\leq n-1,\ j=1,m;\)} \\ -2, & \hbox{\(1\leq i\leq n-1,\ 2\leq j\leq m-1\).} \end{array} \right.$$ $$\varphi((x_n,y_j)(x_{1},y_j))= \left\{ \begin{array}{ll} -1, & \hbox{\(\ j=1,m;\)} \\ -2, & \hbox{\(\ 2\leq j\leq m-1.\)} \end{array} \right.$$ $$\varphi((x_i,y_j)(x_{i},y_{j+1}))= 2,\quad 1\leq i\leq n,\ 1\leq j \leq m-1$$ Now, using this assignment we will prove that the sum of flow at each vertex is zero. For this purpose we will find the weight of each vertex and the weight of a vertex is the sum of all labels of edges adjacent to it. The weight for each vertex is calculated below: \begin{eqnarray*} wt(x_i,y_1) &=& \varphi((x_i,y_1)(x_{i+1},y_1))+\varphi((x_i,y_1)(x_i,y_2))\\ &\ & +\varphi((x_{i-1},y_1)(x_i,y_1)) \\ &=& 0, \quad\quad\quad\quad\quad\quad\quad \textrm{for}\; 2\leq i \leq n-1 \end{eqnarray*} \begin{eqnarray*} wt(x_1,y_1) &=&\varphi((x_1,y_1)(x_2,y_1))+\varphi((x_n,y_1)(x_1,y_1))\\ &\ &+\varphi((x_1,y_1)(x_1,y_2))\\ &=& 0,\\ wt(x_n,y_1) &=&\varphi((x_n,y_1)(x_{n-1},y_1))+\varphi((x_1,y_1)(x_n,y_1))\\ &\ & +\varphi((x_n,y_1)(x_n,y_2))\\ &=& 0,\\ wt(x_i,y_m) &=&\varphi((x_i,y_m)(x_{i+1},y_m))+\varphi((x_i,y_m)(x_i,y_{m-1}))\\ &\ & +\varphi((x_{i-1},y_m)(x_i,y_m))\\ &=& 0, \quad\quad\quad\quad\quad\quad\quad \textrm{for}\; 2\leq i \leq n-1\\ wt(x_1,y_m) &=&\varphi((x_1,y_m)(x_2,y_m))+\varphi((x_n,y_m)(x_1,y_m))\\ &\ & +\varphi((x_1,y_m)(x_1,y_{m-1}))\\ &=& 0,\\ wt(x_n,y_m) &=&\varphi((x_n,y_m)(x_{n-1},y_m))+\varphi((x_n,y_m)(x_1,y_m))\\ &\ & +\varphi((x_n,y_m)(x_n,y_{m-1}))\\ &=& 0. \end{eqnarray*} for \(2\leq j\leq m-1\) and \(2\leq i\leq n-1\) \begin{eqnarray*} wt(x_i,y_j) &=&\varphi((x_i,y_j)(x_{i+1},y_j))+\varphi((x_i,y_j)(x_{i-1},y_j))\\ &\ & +\varphi((x_{i},y_j)(x_i,y_{j+1}))+\varphi((x_{i},y_j)(x_i,y_{j-1}))\\ &=& 0,\\ wt(x_1,y_j) &=&\varphi((x_1,y_j)(x_{2},y_j))+\varphi((x_1,y_j)(x_{n},y_j))\\ &\ &+\varphi((x_{1},y_j)(x_1,y_{j+1}))+\varphi((x_{1},y_j)(x_1,y_{j-1}))\\ &=& 0,\\ wt(x_n,y_j) &=&\varphi((x_n,y_j)(x_{1},y_j))+\varphi((x_n,y_j)(x_{n-1},y_j))\\ &\ &+\varphi((x_{n},y_j)(x_n,y_{j+1}))+\varphi((x_{n},y_j)(x_n,y_{j-1}))\\ &=& 0, \end{eqnarray*} By above computations we can see that \(\varphi\) give us a zero-sum 3-flow. So we get \(F(P_n^m)\leq3\). This concludes the result.

4. Zero-Sum Flow Number of Book graph \((P_n+P_1)\times P_2\)

The join graph \(G + H\) of two graphs \(G\) and \(H\) is their graph union with all the edges that connect the vertices of \(G\) with the vertices of \(H\) . The cartesian product graph \((P_n + P_1) \times P_2\) is a graph with the vertex set \(V((P_n + P_1) \times P_2) = \{u, u_1, u_2\ldots u_n, v, v_1, v_2\ldots v_n\}\) and the edge set \(E((P_n + P_1)\times P_2) = \{ uu_i,\ vv_i,\ u_iv_i|\ i = 1,2\ldots, n\}\cup \{u_iu_{i+1},\ v_iv_{i+1}|\ i = 1\ldots n- 1\} \cup \{uv\}.\)

Theorem 4.1. The zero-sum flow number \(F((P_n + P_1) \times P_2)\) of \((P_n + P_1) \times P_2\) is \(3\) for all \(n\geq3\).

Proof. Note that the degree of vertices \(u_1,\ u_n,\ v_1\) and \(v_n\) is 3, so \(\{-1,1\}\) assignment for the edges will not give us a zero-sum flow therefore \(F((P_n + P_1) \times P_2)\geq3\). Now we will show that \(F((P_n + P_1) \times P_2)\leq3\) and for this purpose we shall consider the labeling \(\varphi:E((P_n + P_1) \times P_2)\rightarrow \{1, -1,-2\}\) on the edges of \((P_n + P_1) \times P_2\) graph. \(\varphi(uv)=\left\{ \begin{array}{ll} -1, & \hbox{if $n$ is odd;} \\ -2, & \hbox{if $n$ is even.} \end{array} \right. \) \(\varphi(uu_i)=\varphi(vv_i)=\left\{ \begin{array}{ll} 1, & \hbox{if $i=1,\ n$;} \\ (-1)^i, & \hbox{$n$ is even, $2\leq i\leq n-1$;} \\ (-1)^{i+1}, & \hbox{$n$ is odd, $2\leq i\leq n-1$.} \end{array} \right.\) \(\varphi(u_iv_i)=\left\{ \begin{array}{ll} -2, & \hbox{if $i=1,\ n$;} \\ (-1)^{i+1}, & \hbox{$n$ is even, $2\leq i\leq n-1$;} \\ -1, & \hbox{$n$ is odd, $i=2$;}\\ (-1)^{i}, & \hbox{$n$ is odd, $3\leq i\leq n-1$;} \end{array} \right.\) \(\varphi(u_iu_{i+1})=\varphi(v_iv_{i+1})=\left\{ \begin{array}{ll} (-1)^{i+1}, & \hbox{$n$ is even, $1\leq i\leq n-1$;} \\ 1, & \hbox{$n$ is odd, $i=1$;} \\ (-1)^{i}, & \hbox{$n$ is odd, $2\leq i\leq n-1$.} \end{array} \right.\) We can see that \(\varphi\) is an edge labeling from \(E((P_n + P_1) \times P_2)\) to \(\{1,-1,-2\}\). Moreover it is easy to check that \begin{eqnarray*} % \nonumber to remove numbering (before each equation) wt(u)&=& \sum_{i=1}^n\varphi(uu_i)+\varphi(uv)=0,\\ wt(v)&=& \sum_{i=1}^n\varphi(vv_i)+\varphi(uv)=0,\\ wt(u_1)&=& \varphi(uu_1)+ \varphi(u_1u_2)+\varphi(u_1v_1)=0,\\ wt(u_n)&=& \varphi(uu_{n})+ \varphi(u_nu_{n-1})+\varphi(u_nv_n)=0,\\ wt(v_1)&=& \varphi(vv_{1})+ \varphi(v_1v_{2})+\varphi(u_1v_1)=0,\\ wt(v_n)&=& \varphi(vv_{n})+ \varphi(v_nv_{n-1})+\varphi(u_nv_n)=0, \end{eqnarray*} For \(2\leq i\leq n-1\), \begin{eqnarray*} % \nonumber to remove numbering (before each equation) wt(u_i)&=& \varphi(uu_{i})+ \varphi(u_iu_{i-1})+\varphi(u_iu_{i+1})+\varphi(u_iv_i)=0 \end{eqnarray*} For \(2\leq i\leq n-1\), \begin{eqnarray*} % \nonumber to remove numbering (before each equation) wt(v_i)&=& \varphi(vv_{i})+ \varphi(v_iv_{i-1})+\varphi(v_iv_{i+1})+\varphi(u_iv_i)=0 \end{eqnarray*} So \(\varphi\) is a zero-sum 3-flow of \((P_n + P_1) \times P_2\). Therefore \(F((P_n + P_1) \times P_2)\leq3\). This conclude the statement of the theorem.

Acknowledgments

This research is supported by the Start-Up Research Grant 2016 of United Arab Emirates University (UAEU), Al Ain, United Arab Emirates via Grant No. G00002233 and UPAR Grant of UAEU via Grant No. G00002590.

Competing Interests

The authors declare that they have no competing interests.

1. Introduction

Let \(G\) be a simple, undirected graph with vertex set \(V(G) = \{v_1, v_2, \ldots, v_n\}\) and edge set \(E(G) = \{e_1, e_2, \ldots, e_m\}\). The degree of a vertex \(v_i\) denoted by \(d_{G}(v_i)\) is the number of edges incident to it. If all vertices have same degree equal to \(r\) then \(G\) is called an \(r\)-regular graph.

The adjacency matrix of \(G\) is a square matrix of order \(n\), defined as \(A=A(G)=[a_{ij}]\), where \(a_{ij}=1\) if \(v_i\) is adjacent to \(v_j\) and \(a_{ij}=0\), otherwise. The characteristic polynomial of \(A(G)\) is denoted by \(\phi(G:\lambda)\), that is, \(\phi(G:\lambda)= \det|\lambda I-A(G)|\), where \(I\) is an identity matrix. The characteristic polynomial of the adjacency matrix of a complete graph \(K_n\) is \(\phi(K_n : \lambda) = (\lambda - n + 1)(\lambda + 1)^{n-1}\). The roots of the equation \(\phi(G:\lambda)=0\) are called the adjacency eigenvalues of \(G\) [1] and they are denoted by \(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\). Two non-isomorphic graphs are said to be cospectral if they have same eigenvalues. For any graph \(G\), \(-\Delta\leq \lambda_{i} \leq \Delta\), where \(\Delta\) is the maximum degree. Thus for an \(r\)-regular graph, \(\lambda_{i}+ r\geq 0\) for \(i= 1, 2, \ldots, n\).

The vertex-edge incidence matrix of \(G\) is defined as \(B=B(G)=[b_{ij}]\), where \(b_{ij}=1\) if the vertex \(v_i\) is incident to an edge \(e_j\) and \(b_{ij}=0\), otherwise.

It is easy to observe that [1] $$BB^T=A+D,$$ where \(D= \text{diag} [d_G(v_1), d_G(v_2), \ldots, d_G(v_n)]\) is a diagonal degree matrix of \(G\) and \(B^T\) is the transpose of \(B\). If \(G\) is an \(r\)-regular graph, then The other matrices of a graph exists in the literature such as distance matrix [2], Laplacian matrix [3], Laplacian distance matrix [4], sum-eccentricity matrix [5, 6], degree sum matrix [7, 8], degree sum adjacency matrix [9], Zagreb matrix [10], degree subtraction matrix [11], degree product matrix [12], degree square sum matrix [13] and average-degree eccentricity matrix [14]. In [15] the degree subtraction adjacency (DSA) matrix is defined as \(DSA(G)=[d_{ij}]\), where \[ d_{ij} = \left\{ \begin{array}{ll} d_{G}(v_i)- d_{G}(v_j), & \text{if $v_i$ is adjacent to $v_j$} \\ 0, & \text{otherwise}. \end{array} \right. \] The characteristic polynomial of \(DSA(G)\) is called the \(DSA\)-polynomial and is denoted by \(\psi(G:\xi)\). Thus \(\psi(G:\xi) = \det(\xi I-DSA(G))\), where \(I\) is an identity matrix of order \(n\). For any regular graph of order \(n\), \(\psi(G:\xi)=\xi^n\). The line graph \(L(G)\) of a regular graph is regular. Hence \(\psi(L(G):\xi)=\xi^m\), where \(m\) is the number of edges of \(G\). The eigenvalues of \(DSA(G)\), denoted by \(\xi_1, \xi_2, \ldots, \xi_n\) are called DSA-eigenvalues of \(G\). Two non-isomorphic graphs are said to be DSA-cospectral if they have same DSA-eigenvalues. Since \(DSA(G)\) is a skew-symmetric matrix, its eigenvalues are purely imaginary or zero. The DSA-energy of a graph \(G\) is defined as The Eq. (2) is analogous to the ordinary graph energy defined as [16] $$E_A(G)=\sum_{i=1}^{n}|\lambda_i|,$$ where \(\lambda_1, \lambda_2, \ldots, \lambda_n\) are the adjacency eigenvalues of \(G\). The ordinary graph energy is well studied by many researchers [17]. In [15] the DSA-polynomial and DSA-energy of a path, complete bipartite graph, wheel, windmill graph and corona graph have been obtained. In this paper we obtain the DSA-eigenvalues and DSA-energy of subdivision graph, semitotal point graph, semitotal line graph and of toal graph of regular graphs.

2. DSA-eigenvalues

A subdivision graph of \(G\) is a graph \(S(G)\) obtained from \(G\) by inserting a new vertex on each edge of \(G\) [18]. Thus if \(G\) has \(n\) vertices and \(m\) edges, then \(S(G)\) has \(n+m\) vertices and \(2m\) edges. If \(u\in V(G)\) then \(d_{S(G)}(u)= d_{G}(u)\) and if \(v\) is subdivided vertex then \(d_{S(G)}(v)= 2\).

Lemma 2.1. If \(M\) is a non-singular matrix, then we have \[ \left| \begin{array}{cc} M & N \\ P & Q \end{array}\right| = |M||Q-PM^{-1}N|. \]

Theorem 2.2. Let \(G\) be an \(r\)-regular graph on \(n\) vertices and \(m\) edges. Then \[ \psi(S(G):\xi) = \left\{\begin{array}{ll} \xi^{\frac{n}{2}}(\xi^2+2)^{\frac{n}{2}}, & \mathrm{if} \hspace{4mm} r= 1 \\[2mm] \xi^{2n}, & \mathrm{if} \hspace{4mm} r = 2\\[2mm] (-1)^{n} (r-2)^{2n} \xi^{m-n} \phi\left(G: \frac{-\xi^2-r(r-2)^{2}}{(r-2)^2}\right), & \mathrm{if} \hspace{4mm} r \geq 3. \end{array} \right. \]

Proof. (i) If \(r=1\), then \(G\) is a union of \(k \geq 1\) edges. Thus \(G\) has \(n=2k\) vertices and \(k\) edges. The vertices of \(S(G)\) can be labeled in such a way that \[ DSA(S(G)) = \left[ \begin{array}{cc} O & B^T \\ -B & O \end{array}\right], \] where \(B\) is vertex-edge incidence matrix of \(G\) and \(O\) is zero matrix. Therefore by Lemma 2.1 and Eq.(1) \begin{eqnarray*} \psi(S(G) : \xi) & = & \left|\begin{array}{cc}\xi I_m & - B^T \\ B & \xi I_n \\ \end{array}\right|\\ & = & \xi^m \left|\xi I_{n} + B \frac{I_{m}}{\xi} B^T \right| \\ & = & \xi^{m-n} \left|\xi^2 I_{n} + BB^T \right| \\ & = & \xi^{m-n} \left|\xi^2 I_{n} + A + rI_{n}\right| \\ & = & (-1)^n\xi^{m-n} \left|-(\xi^2 + r)I_{n} - A \right| \\ & = & (-1)^n\xi^{m-n} \phi(G: -(\xi^2 + r)) \\ & = & (-1)^{2k}\xi^{k-2k} \left((\xi^2+1)^2-1 \right)^{k}\\ & = & \xi^{k} (\xi^2+2)^{k} \\ & = & \xi^\frac{n}{2}(\xi^2+2)^\frac{n}{2}. \end{eqnarray*}

(ii) If \(r=2\), then each component of \(G\) is cycle. Therefore \(S(G)\) is \(2\)-regular graph on \(2n\) vertices. Hence \[ \psi(S(G):\xi)= \xi^{2n}. \]

(iii) Let \(r\geq 3\). The vertices of \(S(G)\) can be labeled in such a way that \[ DSA(S(G)) = \left[ \begin{array}{cc} O & (2-r) B^T \\ -(2-r)B & O \end{array}\right], \] where \(B\) is vertex-edge incidence matrix of \(G\) and \(O\) is zero matrix. Therefore by Lemma 2.1 and Eq.(1) \begin{eqnarray*} \psi(S(G) : \xi) & = & \left|\begin{array}{cc}\xi I_m & - (2-r)B^T \\ (2-r)B & \xi I_n \\ \end{array}\right|\\ & = & \xi^m \left|\xi I_{n} + (2-r)^2 B\frac{I_{m}}{\xi} B^T \right| \\ \nonumber & = & \xi^{m-n} \left|\xi^2 I_{n} + (r-2)^2(A+rI) \right| \\ & = & (-1)^n(r-2)^{2n}\xi^{m-n} \left|\left(\frac{-\xi^2-r(r-2)^2}{(r-2)^2}\right)I_n - A \right| \\ & = & (-1)^n(r-2)^{2n}\xi^{m-n} \phi\left(G: \frac{-\xi^2- r(r-2)^2}{(r-2)^2}\right). \end{eqnarray*}

By Theorem 2.2, we have following corollary.

Corollary 2.3. Let \(G\) be an \(r\)-regular graph on \(n\) vertices and \(m\) edges. Let \(\lambda_1, \lambda_2, \ldots, \lambda_n\) be the adjacency eigenvalues of \(G\) and \(\mathbf{i}=\sqrt{-1}\).

(i) If \(r=1\) then DSA-eigenvalues of \(S(G)\) are \(0\) (\(\frac{n}{2}\) times) and \(\pm \mathbf{i}\sqrt{2}\) (\(\frac{n}{2}\) times).

(ii) If \(r=2\) then DSA-eigenvalues of \(S(G)\) are all zeros.

(iii) If \(r\geq 3\) then DSA-eigenvalues of \(S(G)\) are \(0\) (\(m-n\) times) and \(\pm \mathbf{i}(r-2)\sqrt{r+\lambda_i}\), \(i=1, 2, \ldots, n\).

The semitotal pont graph of \(G\), denoted by \(T_{1}(G)\), is a graph with vertex set \(V(G)\cup E(G)\) and two vertices in \(T_{1}(G)\) are adjacent if they are adjacent vertices in \(G\) or one is a vertex and other is an edge incident to it in \(G\) [19]. Note that if \(u\in V(G)\) then \(d_{T_{1}(G)}(u)= 2 d_{G}(u)\) and if \(e\in E(G)\) then \(d_{T_{1}(G)}(e)= 2\).

Theorem 2.4. Let \(G\) be an \(r\)-regular graph on \(n\) vertices and \(m\) edges. Then \[ \psi(T_{1}(G):\xi) = \left\{\begin{array}{ll} \xi^{m+n}, & \mathrm{if} \hspace{4mm} r = 1\\[2mm] (-1)^n (2r-2)^{2n}\xi^{m-n} \phi\left(G: \frac{-\xi^2- r(2r-2)^2}{(2r-2)^2}\right), & \mathrm{if} \hspace{4mm} r\geq 2. \end{array} \right. \]

Proof. (i) If \(r=1\), then \(T_{1}(G)\) is a regular graph of degree two on \(m+n\) vertices. Hence \[ \psi(T_{1}(G):\xi) = \xi^{m+n}. \]

(ii) Let \(r \geq 2\). The vertices of \(T_1(G)\) can be labeled in such a way that \[ DSA(T_1(G)) = \left[\begin{array}{cc} O & (2-2r) B^T \\ -(2-2r)B & O \end{array}\right], \] where \(B\) is vertex-edge incidence matrix of \(G\) and \(O\) is a zero matrix. Therefore by Lemma 2.1, and Eq. (1) \begin{eqnarray*} \psi(T_1(G) : \xi) & = & \left|\begin{array}{cc}\xi I_m & - (2-2r)B^T \\ (2-2r)B & \xi I_n \\ \end{array}\right|\\ & = & \xi^m \left|\xi I_{n} + (2r-2)^2 B\frac{I_{m}}{\xi} B^T \right| \\ \nonumber & = & \xi^{m-n} \left|\xi^2 I_{n} + (2r-2)^2(A+rI) \right| \\ & = & (-1)^n(2r-2)^{2n}\xi^{m-n} \left|\left(\frac{-\xi^2-r(2r-2)^2}{(2r-2)^2}\right)I_n - A \right| \\ & = & (-1)^n(2r-2)^{2n}\xi^{m-n} \phi\left(G: \frac{-\xi^2- r(2r-2)^2}{(2r-2)^2} \right). \end{eqnarray*}

By Theorem 2.4, we have following corollary.

corollary 2.5. Let \(G\) be an \(r\)-regular graph on \(n\) vertices and \(m\) edges. Let \(\lambda_1, \lambda_2, \ldots, \lambda_n\) be the adjacency eigenvalues of \(G\) and \(\mathbf{i}=\sqrt{-1}\).

(i) If \(r=1\) then DSA-eigenvalues of \(T_1(G)\) are all zeros.

(ii) If \(r\geq 2\) then DSA-eigenvalues of \(T_1(G)\) are \(0\) (\(m-n\) times) and \(\pm \mathbf{i}(2r-2)\sqrt{r+\lambda_i}\), \(i=1, 2, \ldots, n\).

Semitotal line graph of \(G\), denoted by \(T_2(G)\), is a graph with vertex set \(V(G)\cup E(G)\) and two vertices in \(T_2(G)\) are adjacent if one is a vertex and other is an edge incident to it in \(G\) or both are edges adjacent in \(G\) [1]. Note that if \(u\in V(G)\) then \(d_{T_2(G)}(u)= d_{G}(u)\) and if \(e= uv \in E(G)\) then \(d_{T_2(G)}(e)= d_{G}(u)+d_{G}(v)\).

Theorem 2.6. Let \(G\) be an \(r\)-regular graph \((r\geq 1)\) on \(n\) vertices and \(m\) edges. Then \[ \psi(T_2(G) : \xi) = (-1)^n r^{2n}\xi^{m-n} \phi\left(G: \frac{-\xi^2- r^3}{r^2} \right). \]

Proof. The vertices of \(T_2(G)\) can be labeled in such a way that \[ DSA(T_2(G)) = \left[ \begin{array}{cc} O & r B^T \\ - rB & O \end{array}\right], \] where \(B\) is vertex-edge incidence matrix of \(G\) and \(O\) is a zero matrix. Therefore by Lemma 2.1 and Eq. (1) \begin{eqnarray*} \psi(T_2(G) : \xi) & = & \left|\begin{array}{cc}\xi I_m & - rB^T \\ rB & \xi I_n \\ \end{array}\right|\\ & = & \xi^m \left|\xi I_{n} + r^2 B\frac{I_{m}}{\xi} B^T \right| \\ \nonumber & = & \xi^{m-n} \left|\xi^2 I_{n} + r^2(A+rI) \right| \\ & = & (-1)^n r^{2n}\xi^{m-n}\left |\left(\frac{-\xi^2-r^3}{r^2}\right)I_n - A \right| \\ & = & (-1)^n r^{2n}\xi^{m-n} \phi\left(G: \frac{-\xi^2 - r^3}{r^2} \right). \end{eqnarray*}

By Theorem 2.6, we have following corollary.

Corollary 2.7. Let \(G\) be an \(r\)-regular graph \((r\geq 1)\) on \(n\) verties and \(m\) edges. Let \(\lambda_1, \lambda_2, \ldots, \lambda_n\) be the adjacency eigenvalues of \(G\) and \(\mathbf{i}=\sqrt{-1}\). Then DSA-eigenvalues of \(T_2(G)\) are \(0\) (\(m-n\) times) and \(\pm \mathbf{i}r\sqrt{r+\lambda_{i}}\), \(i=1, 2, \ldots, n\).

Total graph of \(G\), denoted by \(T(G)\), is a graph with vertex set \(V(G)\cup E(G)\) and two vertices in \(T(G)\) are adjacent if and only if they are adjacent vertics of \(G\) or adjacent edges of \(G\) or one is a vertex and other is an edge incident to it in \(G\) [18]. Total graph of a regular graph is regular. Hence if \(G\) is regular, then \[ \psi(T(G): \xi) = \xi^{m+n}. \] Two different graphs having same eigenvalues are called cospectral. If \(G_1\) and \(G_2\) are adjacency cospectral graphs with same regularity, then by Corollaries 2.3, 2.5, and 2.7, the graphs \(S(G_1)\) and \(S(G_2)\); \(T_1(G_1)\) and \(T_1(G_2)\); \(T_2(G_1)\) and \(T_2(G_2)\) form a pair of DSA-cospectral graphs.

3. DSA-energy

By Corollaries 2.3, 2.5, and 2.7 and by Eq. (2) we get the following proposition.

Proposition 3.1. Let \(G\) be an \(r\)-regular graph on \(n\) vertices and \(m\) edges. Let \(\lambda_1, \lambda_2, \ldots, \lambda_n\) be the adjacency eigenvalues of \(G\). Then

(i) \[ DSAE(S(G)) = \left\{\begin{array}{ll} n\sqrt{2}, & \mathrm{if} \hspace{4mm} r= 1 \\[2mm] 0, & \mathrm{if} \hspace{4mm} r = 2\\[2mm] 2(r-2)\sum\limits_{i=1}^{n}\sqrt{r+\lambda_{i}}, & \mathrm{if} \hspace{4mm} r \geq 3. \end{array} \right. \]

(ii) \[ DSAE(T_1(G)) = \left\{\begin{array}{ll} 0, & \mathrm{if} \hspace{4mm} r= 1 \\[2mm] 2(2r-2)\sum\limits_{i=1}^{n}\sqrt{r+\lambda_{i}}, & \mathrm{if} \hspace{4mm} r \geq 2. \end{array} \right. \]

(iii) \(DSAE(T_2(G)) = 2r\sum\limits_{i=1}^{n}\sqrt{r+\lambda_{i}}\) for \(r \geq 1\).

(iv) \(DSAE(T(G))= 0\) for \(r\geq 1\).

By Proposition 3.1 we have following result.

Proposition 3.2. If \(G\) is an \(r\)-regular graph \((r\geq 3)\) on \(n\) vertices, then

(i) \((2r-2) DSAE(S(G)) = (r-2) DSAE(T_1(G))\);

(ii) \(r DSAE(S(G)) = (r-2) DSAE(T_2(G))\);

(iii) \(r DSAE(T_1(G)) = (2r-2) DSAE(T_2(G))\).

Proposition 3.3. Let \(G\) be an \(r\)-regular graph \((r\geq 3)\) on \(n\) vertices. Then $$DSAE(S(G)) < DSAE(T_2(G)) < DSAE(T_1(G)).$$

Proof. For \(r\geq 3\), we see that $$r-2 < r < 2r-2.$$ Hence by Proposition 3.2, this implies $$DSAE(S(G)) < DSAE(T_2(G)) < DSAE(T_1(G)).$$

Competing Interests

The authors declare that they have no competing interests.

1. Introduction

Throughout this paper, \(G\) denotes a simple connected undirected graph with vertex set \(V(G)\) and edge set \(E(G)\). Let \(d_G(u)\) and \(N_G(u)\) be the degree and neighbor set of vertex \(u\) in \(G\), respectively. \(n_G(j)\) is the number of the vertices with degree \(j\) in \(G\). For a connected graph \(G\) with \(u\in V(G)\), if \(G-u\) is not connected, then \(u\) is called a cut-vertex of \(G\). Let \(X\) be a subset of \(V(G)\), we use \(G[X]\) to denote the subgraph of \(G\) induced by \(X\).

A cactus graph , or cactus for short, is a connected graph in which no edge lies in more than one cycle. Consequently, each block of a cactus is either an edge or a cycle. A cycle of length \(k\) is denoted by \(C_k\), and \(C_k\) is always called a \(k\)-polygon in the sequel. If each block of a cactus \(G\) is a \(k\)-polygon, then \(G\) is called a \(k\)-polygonal cactus . Hereafter, if there is no risk of confusion, we always call a \(k\)-polygon as a polygon, and we always simplify \(d_G(u)\) and \(N_G(u)\) as \(d(u)\) and \(N(u)\), respectively.

Let \(\mathcal {G}_{n,k}\) be the class of \(k\)-polygonal cacti with \(n\geq3\) blocks. Suppose that \(G\in \mathcal {G}_{n,k}\) . If \(C_k\) contains exactly one cut-vertex, then \(C_k\) is called a pendent polygon . While \(C_k\) is called a non-pendent polygon if \(C_k\) contains at least two cut-vertices.

A cactus chain is a special \(k\)-polygonal cactus graph such that each polygon has at most two cut-vertices, and each cut-vertex is shared by exactly two polygons. When \(G\) is a cactus chain, then the number of polygons is called the length of \(G\). For convenience, we use the notation \(\mathcal{T}_{n,k}\) to denote the class of cactus chains of length \(n\) such that each polygon is a \(k\)-polygon. From the definition, each cactus chain of \(\mathcal{T}_{n,k}\) has exactly \(n-2\) non-pendent polygons and two pendent polygons. When \(k=3\) and \(n\geq 3\), it is easy to see that the cactus chain of \(\mathcal{T}_{n,k}\) is unique. However, when \(k\geq4\) and \(n\geq3\), \(\mathcal{T}_{n,k}\) is not unique.

A star-like cactus \(W_{n,k}\) is a special \(k\)-polygonal cactus graph with \(n\) polygons such that all polygons have a common vertex. From the definition, \(W_{n,k}\) is unique and all polygons of \(W_{n,k}\) are pendent polygons and \(W_{n,k}\) contains exactly one vertex with degree being equalto \(2n\) and the degree of all the other vertices of \(W_{n,k}\) is equal to two.

Among all the vertex-degree-based graph invariants, the first Zagreb index \(M_1(G)\) [1] and zeroth-order Randić index \(R^{0}(G)\) [2] are two famous topological indices, where $$M_{1}(G)=\sum_{u\in V(G)} \big(d(u)\big)^{2},\,\,\text{and}\,\,R^{0}(G)=\sum_{u\in V(G)} \big(d(u)\big)^{-\frac{1}{2}}.$$ In what follows, \(\alpha\) always denotes a real number such that \(\alpha\not\in \{0,1\}\). As a generalization of \(M_1(G)\) and \(R^{0}(G)\), Li and Zheng [3] put forward the concept of first general Zagreb index \(R_{\alpha}^{0}(G)\), where $$ R_{\alpha}^{0}(G)=\sum_{u\in V(G)} \big(d(u)\big)^{\alpha}.$$ From the definition, it is easy to see that \(M_1(G)=R_{2}^{0}(G)\) and \(R^{0}(G)=R_{-\frac{1}{2}}^{0}(G)\). In some literature, \(R_{\alpha}^{0}(G)\) is also called the zeroth-order general Randić index of \(G\) [4, 5, 6].

In what follows, denote by $$\Phi(n,k,\alpha)=\big(n-1\big)4^{\alpha}+\big(nk-2n+2\big)2^{\alpha},\,\,\text{and}\,\,\, \Psi(n,k,\alpha)=\big(2n\big)^{\alpha}+n\big(k-1\big)2^{\alpha}.$$ Recently, the research on zeroth-order general Randić index of cacti had attracted more and more attention. For instance, Ali et al. [4] characterized the extremal polyomino chains with respect to the zeroth-order general Randić index, Hua et al. [6] identified the extremal unicycle graphs with maximum and minimum zeroth-order genenral Randić index and Hu et al. [5] determined the extremal connected \((n,m)\)-graphs with minimum and maximum zeroth-order general Randić index. In this paper, we shall determine the extremal \(k\)-polygonal cactus with \(n\geq 3\) cycles for \(k\geq3\) with respect to the zeroth-order general Randić index, that is,

Theorem 1.1. Let \(G\) be a cactus of \(\mathcal{G}_{n,k}\), where \(n\geq3\), \(k\geq3\) and \(\alpha\) is a real number.

\((i)\) If \(\alpha < 0 \) or \(\alpha > 1\), then \(\Phi(n,k,\alpha)\leq R_{\alpha}^{0}(G)\leq \Psi(n,k,\alpha)\), where the left equality holds if \(G\in\mathcal{T}_{n,k}\) and the right equality holds if and only if \(G \cong W_{n,k}\).

\((ii)\) If \(0< \alpha< 1\), then \(\Psi(n,k,\alpha)\leq R_{\alpha}^{0}(G)\leq \Phi(n,k,\alpha)\), where the left equality holds if and only if \(G \cong W_{n,k}\) and the right equality holds if \(G\in\mathcal{T}_{n,k}\).

Remark 1.2. It is easy to see that \(\mathcal{T}_{n,k}\) is unique for \(k=3\) and \(n\geq3\), but not unique for \(k\geq4\) and \(n\geq3\). By Theorem 1.1, \(R_{\alpha}^{0}(G)=\Phi(n,k,\alpha)\) holds for every cactus \(G\in \mathcal{T}_{n,k}\). Furthermore, the cacti of \(\mathcal{T}_{n,k}\) are not all the extremal cacti of Theorem 1.1, to see this, let \(G_1\) and \(G_2\) be the two cacti as shown in Fig.1. By an elementary computation, we have \(R_{\alpha}^{0}(G_1)=R_{\alpha}^{0}(G_2)=\Phi(4,6,\alpha)\), but \(G_2\not\in \mathcal{T}_{4,6}\).

2. The proof of Theorem 1.1

This section dedicates to the proof of Theorem 1.1.

Lemma 2.1. Let \(f(x)=x^{\alpha}-(x-2)^{\alpha}\). If \(x>2\), then \(f(x)\) is decreasing for \(0< \alpha< 1 \) and increasing for \(\alpha< 0 \) or \(\alpha>1\).

Proof. By Lagrange's mean value theorem, \(f'(x)=\alpha\left(x^{\alpha-1}-(x-2)^{\alpha-1}\right)=2\alpha(\alpha-1)\Theta^{\alpha-2}\), where \(x>2\) and \(x-2< \Theta < x\). It is easy to see that \(f'(x)\) is negative for \(0 < \alpha < 1\) and \(f'(x)\) is positive for \(\alpha < 0 \) or \(\alpha > 1\). Thus, the result holds.

Recall that \(\mathcal{T}_{n,k}\) is the class of cactus chains of length \(n\) such that each polygon is a \(k\)-polygon. From the definition, if \(k=3\) and \(n\geq3\), then \(\mathcal{T}_{n,k}\) is unique. However, when \(k\geq4\) and \(n\geq3\), \(\mathcal{T}_{n,k}\) is not unique. On the other hand, \( W_{n,k}\) is always unique when \(k\geq3\) and \(n\geq3\). The following result implies that \(R_{\alpha}^{0}(G)\) is a constant for either \(G\in \mathcal{T}_{n,k}\) or \(G\cong W_{n,k}\).

Lemma 2.2. Let \(k\geq3\) and \(n\geq1\) be two integers. \((i)\) If \(G\in \mathcal{T}_{n,k}\), then \(R_{\alpha}^{0}(G)=\big(n-1\big)4^{\alpha}+\big(nk-2n+2\big)2^{\alpha}\). \((ii)\) If \(G\cong W_{n,k}\), then \(R_{\alpha}^{0}(G)=\big(2n\big)^{\alpha}+n\big(k-1\big)2^{\alpha}\).

Proof. \((i)\) If \(G\in \mathcal{T}_{n,k}\), then \(n_G(4)=n-1\) and \(n_G(2)=nk-2n+2\). Thus, we have $$R_{\alpha}^{0}(G)=\sum_{u\in V(G)} (d(u))^{\alpha}=\big(n-1\big)4^{\alpha}+\big(nk-2n+2\big)2^{\alpha}.$$

\((ii)\) If \(G\cong W_{n,k}\), then \(n_G(2n)=1\) and \(n_G(2)=n(k-1)\). Thus, we have $$ R_{\alpha}^{0}(G)=\sum_{u\in V(G)} (d(u))^{\alpha}=\big(2n\big)^{\alpha}+n\big(k-1\big)2^{\alpha}.$$ This completes the proof of this result.

To prove our main results, we need to introduce more definitions, which were raised in [7]:
Suppose that \(G\in \mathcal{G}_{n,k}\) and \(C^{(1)}_{k}\), \(C^{(2)}_{k}\), \(\ldots,\) \(C^{(s)}_{k}\) are \(s\) cycles of length \(k\) in \(G\), where \(k\geq 3\), \(s\geq 1\) and \(n\geq3\). Let \(V_1=V\left(C^{(1)}_{k}\right)\cup V\left(C^{(2)}_{k}\right)\cup \cdots \cup V\left(C^{(s)}_{k}\right)\) and let \(u_1\) be a cut-vertex of \(C^{(1)}_{k}\) in \(G\) such that \(u_1\) is not a cut-vertex of \(G\big[V_1\big]\). If \(G\big[V_1\big]\) is a cactus chain and each \(k\)-polygon of \(\left\{C^{(1)}_{k}, C^{(2)}_{k}, \ldots, C^{(s)}_{k}\right\}\) has at most two cut-vertices in \(G\), \(C^{(s)}_{k}\) is a pendent polygon of \(G\), the degree of each vertex of \(V_1\setminus\{u_{1}\}\) is at most four in \(G\), then \(G\big[V_1\big]\) is called a pendent cactus chain of length \(s\) of \(G\). Furthermore, if \(G\big[V_1\big]\) is a pendent cactus chain of length \(s\geq 2\), then \(C^{(s-1)}_{k}\) is called a neighbor polygon of the pendent cactus chain. Hereafter, we denote \(L_{s,k}\) as a pendent cactus chain of length \(s\) in a \(k\)-polygonal cactus. From the definition, if \(G\big[V_1\big]\) is a pendent cactus chain of length \(s\geq 2\), then for \(1\leq i\leq s-1\) and \(2\leq j\leq s-1\), each \(C^{(i)}_k\) contains exactly two cut-vertices in \(G\) and the degree of every cut-vertex of \(C^{(j)}_k\) is equal to four in \(G\).

Definition 2.3. Let \(G\) be a cactus of \(\mathcal{G}_{n,k}\) and let \(C^{(1)}_{k}\), \(C^{(2)}_{k}\), \(\ldots,\) \(C^{(s+t)}_{k}\) be \(s+t\) cycles of length \(k\) of \(G\) such that \(G\left[V\left(C^{(1)}_{k}\right)\cup V\left(C^{(2)}_{k}\right)\cup \cdots \cup V\left(C^{(s)}_{k}\right)\right]\) and \(G\Big[V\left(C^{(s+1)}_{k}\right)\cup V\left(C^{(s+2)}_{k}\right)\cup \cdots \cup V\left(C^{(s+t)}_{k}\right)\Big]\) are two pendent cactus chains of length \(s\geq 1\) and \(t\geq 1\), respectively.

\((i)\) If \(u_0\in V\left(C^{(1)}_{k}\right)\cap V\left(C^{(s+1)}_{k}\right)\) and \(d_G(u_0)\geq 6\), then \(u_0\) is called a singular vertex of \(G\).

\((ii)\) If \(C^{(0)}_k\) is a \(k\)-polygon of \(G\) with at least three cut vertices in \(G\) such that \(V\left(C^{(1)}_{k}\right)\cap V\left(C^{(0)}_{k}\right)=\{v_0\}\) and \(V\left(C^{(s+1)}_{k}\right)\cap V\left(C^{(0)}_{k}\right)=\{w_0\}\) with \(d_G(w_0)=d_G(v_0)=4\), then \(C^{(0)}_k\) is called a special polygon of \(G\).

Lemma 2.4. Let \(G\) be a cactus of \(\mathcal{G}_{n,k}\), where \(k\geq 3\) and \(n\geq3\). If \(G\) contains a singular vertex, then \(R_{\alpha}^{0}(G)\) is neither minimum for \(\alpha< 0\) or \(\alpha>1\) and not maximum for \(0 < \alpha< 1\) in \( \mathcal{G}_{n,k}\).

Proof. By contradiction, we assume that \(R_{\alpha}^{0}(G)\) is minimum for \(\alpha< 0\) or \(\alpha>1\) and maximum for \(0< \alpha< 1\) in \( \mathcal{G}_{n,k}\). Let \( u_0 \) be a singular vertex of \(G\) with \(d_G(u_0)=2r\), where \(r\geq 3\). For convenience, we suppose that \(u_0\) is a common vertex of two pendent cactus chains \(L_{t,k}\) and \(L_{s,k}\) in \(G\), where \(s\geq t\geq 1\). Suppose that \(C^{(t)}_k=u_1u_2\cdots u_ku_1\) and \(C^{(s)}_k=w_1w_2\cdots w_kw_1\) are the pendent polygons of \(L_{t,k}\) and \(L_{s,k}\), respectively, such that \(u_1\) and \(w_1\) are two cut-vertices of \(G\). Let \(G'=G-u_1u_2-u_1u_k+w_2u_2+w_2u_k\). By the definition of \(G'\), it it easy to see that

Observation 1. If \(t\geq 2\), then \(u_0\) is also a singular vertex of \(G'\) such that \(u_0\) is a common vertex of two pendent cactus chains \(L_{t-1,k}\) and \(L_{s+1,k}\) in \(G'\).
We consider the following two cases:

Case 1. \(t=1\).
From the definition, we have \begin{eqnarray*} \ R_{\alpha}^{0}(G)-R_{\alpha}^{0}(G')=(2r)^\alpha+2^{\alpha}-(2r-2)^\alpha-4^{\alpha} =(2r)^\alpha-(2r-2)^\alpha-(4^{\alpha}-2^{\alpha}). \end{eqnarray*} By lemma 2.1, since \(2r\geq6>4\), it is easy to see that \(R_{\alpha}^{0}(G)>R_{\alpha}^{0}(G')\) for \(\alpha< 0\) or \(\alpha>1\) and \(R_{\alpha}^{0}(G)< R_{ \alpha}^{0}(G') \) for \(0< \alpha< 1\). No matter which case happens, we can reach a contradiction.

Case 2. \(t\geq2\).
If \(t\geq2\), then from the definition, we have \begin{eqnarray*} \ R_{\alpha}^{0}(G)-R_{\alpha}^{0}(G')&=&4^{\alpha}+2^{\alpha}-2^{\alpha}-4^{\alpha}=0 \end{eqnarray*} Now, by Observation 1 and above equality, there exists a cactus \(G'\) of \(\mathcal{G}_{n,k}\) such that \(R_{\alpha}^{0}(G)=R_{\alpha}^{0}(G')\), \(u_0\) is also a singular vertex of \(G'\) and \(u_0\) is a common vertex of two pendent cactus chains \(L_{t-1,k}\) and \(L_{s+1,k}\) in \(G'\). By repeating the above process, we can conclude that there exists a cactus \(G_1\) of \(\mathcal{G}_{n,k}\) such that \(R_{\alpha}^{0}(G)=R_{\alpha}^{0}(G_1)\), \(u_0\) is also a singular vertex of \(G_1\) and \(u_0\) is a common vertex of two pendent cactus chains \(L_{1,k}\) and \(L_{s+t-1,k}\) in \(G_1\). Now, from the above arguments and Case 1, we can conclude that there exists cactus \(G_0\) of \(\mathcal{G}_{n,k}\) such that \(R_{\alpha}^{0}(G)>R_{\alpha}^{0}(G_0)\) for \(\alpha< 0\) or \(\alpha >1\) and \(R_{\alpha}^{0}(G)< R_{\alpha}^{0}(G_0)\) for \(0< \alpha< 1 \), and \(G_0\) contains no singular vertex, a contradiction. Thus, the result holds.

Lemma 2.5. Let \(G\) be a cactus of \(\mathcal{G}_{n,k}\), where \(n\geq4\) and \(k\geq 3\). If \(G\) contains a special polygon, then there exists \(G_0\in \mathcal{G}_{n,k}\) such that \(R_{\alpha}^{0}(G_0)\leq R_{\alpha}^{0}(G)\) for \(\alpha< 0\) or \(\alpha>1\) and \(R_{\alpha}^{0}(G_0)\geq R_{\alpha}^{0}(G)\) for \(0< \alpha< 1\) and \(G_0\) contains no special polygon.

Proof. Let \(C^{(0)}_k\) be a special polygon, and let \(L_{t,k}\) and \(L_{s,k}\) be two pendent cactus chains of \(G\) such that \(V\big(L_{t,k}\big)\cap V\left(C^{(0)}_k\right)=\{u_0\}\) and \(V\big(L_{s,k}\big)\cap V\left(C^{(0)}_k\right)=\{w_0\}\), where \(s\geq t\geq 1\). Suppose that \(C^{(t)}_k=u_1u_2\cdots u_ku_1\) and \(C^{(s)}_k=w_1w_2\cdots w_kw_1\) are the pendent polygons of \(L_{t,k}\) and \(L_{s,k}\), respectively, such that \(u_1\) and \(w_1\) are two cut-vertices of \(G\). Let \(G'=G-u_1u_2-u_1u_k+w_2u_2+w_2u_k\). By the definition of \(G'\), it it easy to see that

Observation 1. If \(t\geq 2\), then \(C^{(0)}_k\) is also a special polygon of \(G'\) and that \(L_{t-1,k}\) and \(L_{s+1,k}\) are two pendent cactus chains of \(G'\) such that \(V\big(L_{t-1,k}\big)\cap V\left(C^{(0)}_k\right)=\{u_0\}\) and \(V\big(L_{s+1,k}\big)\cap V\left(C^{(0)}_k\right)=\{w_0\}\). We consider all cases as follows, by the definition of \(G'\), we have

Apparently, if \(t\geq 2\), by observation 1 we can conclude that there exists a cactus \(G'\) of \(\mathcal{G}_{n,k}\) such that \(R_{\alpha}^{0}(G)=R_{\alpha}^{0}(G')\), where \(C^{(0)}_k\) is also a special polygon of \(G'\) such that \(L_{t-1,k}\) and \(L_{s+1,k}\) are two pendent cactus chains of \(G'\), \(V\big(L_{t-1,k}\big)\cap V\left(C^{(0)}_k\right)=\{u_0\}\) and \(V\big(L_{s+1,k}\big)\cap V\left(C^{(0)}_k\right)=\{w_0\}\). By repeating the above process, we can also conclude that there exists a cactus \(G_1\) of \(\mathcal{G}_{n,k}\) such that \(R_{\alpha}^{0}(G)=R_{\alpha}^{0}(G_1)\), where \(C^{(0)}_k\) is also a special polygon of \(G_1\) such that \(L_{1,k}\) and \(L_{s+t-1,k}\) are two pendent cactus chains of \(G_1\), \(V\big(L_{1,k}\big)\cap V\left(C^{(0)}_k\right)=\{u_0\}\) and \(V\big(L_{s+t-1,k}\big)\cap V\left(C^{(0)}_k\right)=\{w_0\}\). And now for \(t=1\), through the operation illustrated before and (1), we can construct the corresponding graph \(G_2\) such that \(G_2\in \mathcal{G}_{n,k}\), \(R_{\alpha}^{0}(G)=R_{\alpha}^{0}(G_2)\) and one pendent chain will disappear in \(G_2\).

By repeating the above arguments, we can conclude that there exists \(G_0\in \mathcal{G}_{n,k}\) such that \(R_{\alpha}^{0}(G_0)\leq R_{\alpha}^{0}(G)\) for \(\alpha< 0\) or \(\alpha>1\) and \(R_{\alpha}^{0}(G_0)\geq R_{\alpha}^{0}(G)\) for \(0< \alpha< 1\) and \(G_0\) contains no special polygon for \(k\geq 3\). Thus, the result holds.

Lemma 2.6. [7] Let \(G\) be a cactus of \(\mathcal{G}_{n,k}\), where \(k\geq 3\) and \(n\geq 3\). If \(G\) contains neither singular vertex nor special polygon, then \(G\) must be a cactus chain.

Lemma 2.7. Let \(G\) be a cactus of \(\mathcal{G}_{n,k}\). If \(k\geq 3\) and \(n\geq 3\), then \(R_{\alpha}^{0}(G)\leq \Psi(n,k,\alpha)\) for \(\alpha< 0\) or \(\alpha>1\) and \(R_{\alpha}^{0}(G)\geq \Psi(n,k,\alpha)\) for \(0< \alpha< 1\), where either equality holds if and only if \(G \cong W_{n,k}\).

Proof. Let \(G\) be a cactus of \(\mathcal{G}_{n,k}\) such that \(G\) is an extremal graph of \(\mathcal{G}_{n,k}\), namely, \(R_{\alpha}^{0}(G)\) is as large as possible for \(\alpha< 0\) or \(\alpha>1\), and \(R_{\alpha}^{0}(G)\) is as small as possible for \(0< \alpha< 1 \). We suppose that the degree of vertex \(u_{0}\) is largest among all vertices in \(G\) and \(d_{G}(u_{0})=2r_{1}\). If \(2r_{1}=2n\), then \(G\cong W_{n,k}\), and hence the result already holds. Otherwise, \(2r_{1}< 2n \). Furthermore, we suppose that \(C_{k}^{(1)}\) is a pendent polygon with \(u_{1}\) being its cut-vertex such that \(N(u_{1})\cap V(C_{k}^{(1)})=\{w_{1},w_{k}\}\) and \(d_{G}(u_{1})=2r_{2}\), where \(u_{1}\neq u_0\). Then it is easy to see that \(2\leq r_{2}\leq r_{1}\leq n\). Now, we let \(G_{1}=G-u_1w_1-u_1w_k+u_0w_1+u_0w_k\). By an elementary computation, it follows that \begin{align*}R_{\alpha}^{0}(G)-R_{\alpha}^{0}(G_1)&=(2r_1)^{\alpha}+(2r_2)^{\alpha}-(2r_1+2)^{\alpha}-(2r_2-2)^{\alpha} \\&=\left(2r_2\right)^{\alpha}-\left(2r_2-2\right)^{\alpha}-\big((2r_1+2)^{\alpha}-(2r_1)^{\alpha}\big).\end{align*} Since \(2r_1\geq2r_2\geq4\), by lemma 2.1 we have \(R_{\alpha}^{0}(G)< R_{\alpha}^{0}(G_1)\) for \(\alpha< 0\) or \(\alpha>1\), and \(R_{\alpha}^{0}(G)>R_{\alpha}^{0}(G_1)\) for \(0< \alpha< 1\), which is contrary with the choice of \(G\). Thus, \(u_0\) is the cut-vertex of any pendent polygon. Since \(G\) is a cactus in \(\mathcal{G}_{n,k}\), we have \(G\cong W_{n,k}.\)

Next, we turn to prove Theorem 1.1.

Proof. By Lemma 2.2, \(R_{\alpha}^{0}(G)=\Phi(n,k,\alpha)\) holds for \(G\in \mathcal{T}_{n,k}\), and \(R_{\alpha}^{0}(G)=\Psi(n,k,\alpha)\) holds for \(G\cong W_{n,k}\). Now, we consider the following two cases:

Case 1. \(\alpha< 0\) or \(\alpha>1\).
Then, Lemmas 2.4--2.6 imply that \(R_{\alpha}^{0}(G)\) is minimum if \(G\in \mathcal{T}_{n,k}\). Combining this with Lemma 2.7, we can conclude that \(R_{\alpha}^{0}(G)\) is maximum if and only if \(G\cong W_{n,k}\). Thus, \((i)\) holds.

Case 1. \(0< \alpha< 1\).
By Lemmas 2.4--2.6, \(R_{\alpha}^{0}(G)\) is maximum if \(G\in \mathcal{T}_{n,k}\). Taking Lemma 2.7 into consideration, we can conclude that \(R_{\alpha}^{0}(G)\) is minimum if and only if \(G\cong W_{n,k}\). Thus, \((ii)\) also holds.

Acknowledgments

The authors would like to thank Professor Muhuo Liu for his valuable comments which lead to an improvement of the original manuscript. This paper is supported by Guangdong Province Ordinary University Characteristic Innovation Project (No.2017KTSCX020) and National Undergraduate Training Programs for Innovation and Entrepreneurship (No. 201810564014).

Competing Interests

The authors declare that they have no competing interests.