On characteristic polynomial and energy of Sombor matrix

1. Introduction

The Sombor index \(SO\) is a recently introduced vertex-degree-based topological index [1]. It promptly attracted much attention and its mathematical properties and chemical applications became a topic of a remarkably large number of studies, e.g., [2,3,4,5,6,7,8,9]. Also promptly, the concept of Sombor index was extended to linear algebra, by defining the Sombor matrix, which then led to the investigation of its spectrum and various spectrum-based properties [10,11,12,13,14]. In particular, the energy of the Sombor matrix was much examined [11,12,13,14]. In the present paper we report a few additional results on this matter, with emphasis on the characteristic polynomial and energy.

In this paper, we considered simple, finite, undirected, and connected graphs. Let \(G\) be such a graph, with vertex set \(\mathbf V(G)\) and edge set \(\mathbf E(G)\). If two vertices have a common edge then they are said to be adjacent. If the vertices \(u\) and \(v\) are adjacent, then the edge connecting them is denoted by \(uv\). The number of edges incident to a vertex \(v\) is called the degree of that vertex \(v\), and is denoted by \(d_v\).

In the mathematical and chemical literature, a great number of vertex-degree-based graph invariants of the form

have been considered, where \(\varphi\) is a suitably chosen function, with property \(\varphi(x,y)=\varphi(y,x)\). These invariants are usually referred to as topological indices. Among them are the forgotten topological index [15] \[ F(G)=\sum_{uv\in \mathbf E(G)} \big( d_u^2+d_v^2 \big) = \sum_{u\in \mathbf V(G)} d_u^3, \] the Sombor index [1] \[ SO(G) = \sum_{uv\in \mathbf E(G)} \sqrt{ d_u^2+d_v^2}, \] and many other [16,17].

The adjacency matrix \(\mathbf A(G)=(a_{ij})_{n \times n}\) of the graph \(G\) with vertex set \(\mathbf V(G)=\{v_1,v_2,\dots,v_n\}\), is the symmetric matrix of order \(n\), whose elements are defined as [18]:

The characteristic polynomial of \(\mathbf A(G)\) is \(\phi(G,\lambda) = \det \big[\lambda\,\mathbf I_n - \mathbf A(G) \big]\), where \(\mathbf I_n\) is the unit matrix of order \(n\) [18]. The eigenvalues \(\lambda_1,\lambda_2,\ldots,\lambda_n\) of \(\mathbf A(G)\) form the spectrum of the graph \(G\) [18]. Recall that these eigenvalues coincide with the zeros of \(\phi(G,\lambda)\).

The energy of the graph \(G\) is defined as [19]:

\[ En(G) = \sum_{i=1}^n |\lambda_i|\,. \] The theory of graph spectra, including the theory of graph energy, is nowadays a well elaborated part of discrete mathematics. In parallel with the above specified graph-spectral concepts, we now introduce their Sombor-index-related counterparts. The following definition is an application to the Sombor index of the general spectral theory of matrices associated with vertex-degree-based topological indices of the form (1) [20,21,22].

Definition 1.

• (1) The Sombor matrix \(\mathbf A_{SO}(G)=(so_{ij})_{n \times n}\) of the graph \(G\) with vertex set \(\mathbf V(G)=\{v_1,v_2,\dots,v_n\}\), is the symmetric matrix of order \(n\), whose elements are
  \begin{equation} \label{eq3} so_{ij} = \left\{ \begin{array}{ccl} \sqrt{d_{v_i}^2+d_{v_j}^2} & \hspace{5mm} & \mbox{if \(v_iv_j \in \mathbf E(G)\)} \\ 0 && \mbox{if \(v_iv_j \not \in \mathbf E(G)\)} \\ 0 && \mbox{if \(i=j\)}\,. \end{array} \right. \end{equation}

  (3)
• (2) The Sombor characteristic polynomial of the graph \(G\) is \(\phi_{SO}(G,\lambda) = \det \big[\lambda\,\mathbf I_n - \mathbf A_{SO}(G) \big]\). We will write it in the form \[ \phi_{SO}(G,\lambda) = \sum_{k \geq 0} so(G,k)\,\lambda^{n-k}\,. \]
• (3) The eigenvalues \(\sigma_1,\sigma_2,\ldots,\sigma_n\) of the Sombor matrix \(\mathbf A_{SO}(G)\) form the Sombor spectrum of the graph \(G\).
• (4) The Sombor energy of the graph \(G\) is
  \begin{equation} \label{eq6} En_{SO}(G) = \sum_{i=1}^n |\sigma_i|\,. \end{equation}

  (4)

Since \(\mathbf A_{SO}(G)\) is a real symmetric matrix, all its eigenvalues, i.e., all roots of \(\phi_{SO}(G,\lambda)=0\), are real. Thus, they can be arranged as \(\sigma_1 \geq \sigma_2 \geq \cdots \geq \sigma_n\).

Remark 1. Comparing Equations (2) and (3), we see that the Sombor matrix can be viewed as the ordinary adjacency matrix of a graph with weighted edges, such that the weight of the edge \(v_iv_j\) is \(\sqrt{d_{v_i}^2+d_{v_j}^2}\). This observation allows us to apply to the Sombor matrix and its spectrum the standard methods of graph spectral theory [18], in particular the Sachs coefficient theorem [23].

2. Preliminaries

The following elementary spectral properties of the Sombor matrix were recognized in several earlier studies [10,11,12,13,14].

Lemma 1. Let \(G\) be a graph with Sombor eigenvalues \(\sigma_1, \sigma_2,\ldots,\sigma_n\). Then \begin{align*} \sum_{i=1}^n \sigma_i = & 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \nonumber \end{align*}

or, equivalently, \begin{align*} so(G,1) & = 0 ,\\ so(G,2) & = -F(G), \\ so(G,3) & = -2\,\sum_{\Delta} \prod_{uv\in \mathbf E(\Delta)} \sqrt{d_u^2+d_v^2}, \end{align*} where \(\sum_\Delta\) indicates summation over all triangles contained in the graph \(G\).

Formula (6) can be generalized as follows:

Lemma 2. Let \(p\) be the size of smallest odd cycle contained in the graph \(G\), and let \(\sum_{C_p}\) indicate summation over all cycles of size \(p\) contained in \(G\). Then for \(q=1,3,\ldots,p-2\),

whereas \[ \sum_{i=1}^n \sigma_i^p = 2p\,\sum_{C_p} \prod_{uv \in \mathbf E(C_p)} \sqrt{d_u^2+d_v^2} \] or, equivalently, \[ so(G,p) = -2\,\sum_{C_p} \prod_{uv \in \mathbf E(C_p)} \sqrt{d_u^2+d_v^2}\,. \] If \(G\) does not possess odd cycles, i.e., if \(G\) is bipartite, then relations (7) and \(so(G,q)=0\) hold for all odd values of \(q\).

Proof. Take into account Remark 1, and use the analogous result for ordinary graphs [18].

Lemma 3. [24,25] Suppose that \(a_i\) and \(b_i\) are non negative real numbers for \(1\leq i \leq n\). Then, \[ \left( \sum_{i=1}^n a_i^2\right) \left( \sum_{i=1}^n b_i^2\right) \leq \dfrac{1}{4}\left( \sqrt{\dfrac{M_1\,M_2}{m_1\,m_2}} + \sqrt{\dfrac{m_1\,m_2}{M_1\,M_2}}\right)^2 \left(\sum_{i=1}^n a_i\,b_i \right)^2 \] where \(M_1=\max\limits_{1\leq i \leq n} a_i\)  ,\(M_2=\max\limits_{1\leq i \leq n} b_i\),   \(m_1=\min\limits_{1\leq i \leq n}a_i\) ,  and \(m_2=\min\limits_{1\leq i \leq n}b_i\) .

Lemma 4. [24,25] Using the same notation as in Lemma 3, \[ \left( \sum_{i=1}^n a_i^2\right) \left( \sum_{i=1}^n b_i^2\right) - \left(\sum_{i=1}^n a_i\,b_i \right)^2 \leq \dfrac{n^2}{4}\left( M_1\,M_2-m_1\,m_2\right)^2\,. \]

3. New bounds for Sombor energy

Various lower and upper bounds for Sombor energy were already reported in [10,11,12,13]. In this section we establish a few more.

We first recall a result by Lin and Miao [13], that can be stated in terms of traces of the Sombor matrix. It should be compared with the below Theorem 2. The upper bound was obtained also in [12]. Note that \(tr(\mathbf A_{SO}(G)^2)=2F(G)\) follows from Equation (5).

Theorem 1. [13] Denote the trace of a square matrix \(\mathbf M\) by \(tr(\mathbf M)\). Let \(G\) be a graph on \(n\) vertices. Then \[ \sqrt{tr(\mathbf A_{SO}(G)^2)} \leq En_{SO}(G) \leq \sqrt{n\,tr(\mathbf A_{SO}(G)^2)} \] i.e., \[ \sqrt{2F(G)} \leq En_{SO}(G) \leq \sqrt{2n\,F(G)}\,. \]

Theorem 2. Let \(G\) be a non-trivial graph. Then \[ En_{SO}(G) \geq \sqrt{\dfrac{\big[tr(\mathbf A_{SO}(G)^2)\big]^3}{tr(\mathbf A_{SO}(G)^4)}}\,. \]

Proof. By the Hölder inequality, \[ \sum_{i=1}^n a_i\,b_i \leq \left( \sum_{i=1}^n a_i^p \right)^{1/p} \left( \sum_{i=1}^n b_i^q \right)^{1/q} \] where, \(a_i,b_i \in \mathbf{R}^+\), \((i=1,2,3\dots,n)\). Setting \(a_i=|\sigma_i|^{2/3}\), \(b_i=|\sigma_i|^{4/3}\), \(p=3/2\), and \(q=3\), we get \[ \sum_{i=1}^n |\sigma_i|^2 \leq \left( \sum_{i=1}^n |\sigma_i| \right)^{2/3} \left(\sum_{i=1}^n |\sigma_i|^4 \right)^{1/3} \] which by Equation (4), and bearing in mind that since \(G\) is not an empty graph and thus \(\sum_{i=1}^n|\sigma_i|^4\neq0\), yields Theorem 2.

Theorem 3. If \(\sigma_1\) is the greatest Sombor eigenvalue, then \(En_{SO}(G) \leq 2\sigma_1\). For connected graphs, equality holds if and only if \(G\) is a complete bipartite graph.

Proof. Bearing in mind Equation (4), \[ En_{SO}(G) = |\sigma_1|+\sum_{i=2}^n |\sigma_i| \geq |\sigma_1|+ \left| \sum_{i=2}^n \sigma_i\right|\,. \] On the other hand, \[ \sum_{i=1}^n \sigma_i=0 \ \ \ \ \implies \ \ \ \ \sigma_1=-\sum_{i=2}^n \sigma_i \ \ \ \ \text{and so} \ \ \ \ |\sigma_1|=\left| \sum_{i=2}^n \sigma_i\right|\,. \] Equality holds if \(\sigma_1\) and \(\sigma_n\) are the only non-zero eigenvalues. In view of Remark 1, this happens only if \(G\) is a complete bipartite graph.

Theorem 4. Let \(G\) be a graph with \(n\) vertices. If no Sombor eigenvalue of \(G\) is equal to zero, then \[ En_{SO}(G) \geq \sqrt{\dfrac{8n\,F(G)\,\sigma_\ell\,\sigma_s}{|\sigma_\ell|+|\sigma_s|}} \] where, \(|\sigma_\ell|\) and \(|\sigma_s|\) are, respectively, the largest and smallest absolute values of the eigenvalues in the Sombor spectrum of \(G\). Of course, \(|\sigma_\ell|=\sigma_1\).

Proof. Setting in Lemma 3, \(a_i=|\sigma_i|\) and \(b_i=1\) for \(1\leq i \leq n\), we get \[ \left(\sum_{i=1}^n |\sigma_i|^2 \right) \left( \sum_{i=1}^n 1\right) \leq \dfrac{1}{4}\left( \sqrt{\dfrac{|\sigma_\ell|}{|\sigma_s|}} + \sqrt{\dfrac{|\sigma_s|}{|\sigma_\ell|}}\right)^2 \left(\sum_{i=1}^n |\sigma_i| \right)^2, \] where \(|\sigma_\ell| = \max\limits_{1\leq i \leq n}\{|\sigma_i|\}\) and \(|\sigma_s| = \min\limits_{1\leq i \leq n}\{|\sigma_i|\}\). Then \[ 2F(G)\,n \leq \dfrac{1}{4}\left( \sqrt{\dfrac{|\sigma_\ell|}{|\sigma_s|}} + \sqrt{\dfrac{|\sigma_s|}{|\sigma_\ell|}}\right)^2 \left(\sum_{i=1}^n |\sigma_i| \right)^2 \] and thus \[ \sqrt{8n\,F(G)} \leq \left(\dfrac{|\sigma_\ell|+|\sigma_s|}{\sqrt{\sigma_\ell\,\sigma_s}} \right) En_{SO}(G) \] which straightforwardly leads to Theorem 4.

Theorem 5. Let \(G\) be a connected graph with \(n\) vertices, and \(\sigma_\ell\,,\,\sigma_s\) same as in Theorem 4. Then \[ En_{SO}(G)\geq \sqrt{2n\,F(G)-\dfrac{n^2}{4}(|\sigma_\ell|-|\sigma_s|)} \]

Proof. Setting in Lemma 4, \(a_i=|\sigma_i|\) and \(b_i=1\) for \(1\leq i \leq n\), we get \[ \left( \sum_{i=1}^{n} |\sigma_i|^2\right) \left( \sum_{i=1}^{n} 1\right)-\left(\sum_{i=1}^{n} |\sigma_i| \right)^2 \leq \dfrac{n^2}{4}\left( |\sigma_\ell|-|\sigma_s|\right)^2, \] implying \[ 2F(G)\,n -En_{SO}(G)^2 \leq \dfrac{n^2}{4}(|\sigma_\ell|-|\sigma_s|)^2\,. \] Theorem 5 follows.

4. On Sombor energy of trees

In this section we focus our attention to trees. Let \(T\) be a tree on \(n\) vertices, \(n \geq 2\). The main result in the spectral theory of trees is the formula [18,26,27] where \(m(T,k)\) stands for the number of \(k\)-matchings (= selections of \(k\) mutually independent edges) in the tree \(T\). By definition, \(m(T,1)=n-1\).

As explained in Remark 1, the matrix \(\mathbf A_{SO}(G)\) can be viewed as the adjacency matrix of a graph with weighted edges. This, of course, applies also to trees.

According to the Sachs coefficient theorem [18,23], for the Sombor characteristic polynomial of a tree \(T\), an expression analogous to Equation (8) would hold, namely

The coefficient \(m_{SO}(T,k)\) is equal to the sum of weights coming from all \(k\)-matchings of \(T\). Each particular \(k\)-matching contributes to \(m_{SO}(T,k)\) by the product of the squares of the terms \(\sqrt{d_u^2+d_v^2}\), pertaining to the edges contained in that matching [23]. Thus, let \(M\) be a distinct \(k\)-matching of \(T\), and let \(\mathcal M(k)\) be the set of all such \(k\)-matchings. Then for \(k \geq 1\), \(\mathcal M(k)\) consists of \(m(T,k)\) elements, i.e., \(|\mathcal M(k)| = m(T,k)\).

The weight of a single matching \(M\) is equal to \(\prod\limits_{uv \in M} \big(d_u^2+d_v^2 \big)\) and therefore

provided \(\mathcal M(k) \neq \emptyset\). If, on the other hand, \(\mathcal M(k) = \emptyset\), then \(m_{SO}(T,k)=0\).

We thus see that the coefficients \(m_{SO}(T,k)\) are positive if \(m(T,k)>0\) and are equal to zero if \(m(T,k)=0\). This implies:

Theorem 6. The inertia of the Sombor matrix and of the ordinary adjacency matrix of any tree coincide.

The energy of a tree can be computed from its matching polynomial as [28]: The analogous expression for the Sombor energy is and can be obtained in the exactly same manner as Equation (11) [28,29].

Since, evidently, \(m_{SO}(T,k) > m(T,k)\) holds whenever the tree \(T\) has at least one \(k\)-matching, by comparing Equations (11) and (12), we immediately arrive at:

Theorem 7. For any tree \(T\), \(En_{SO}(T) > En(T)\).

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest

''The authors declare no conflict of interest.''