OMS-Vol. 3 (2019), Issue 1, pp. 417 – 432 Open Access Full-Text PDF

Abstract: Since the emergence of the avian influenza A(H7N9) in the year 2013 in China, several researches have been carried out to investigate the spread. In this paper, a mathematical model describing the transmission dynamics of avian influenza A(H7N9) between human and poultry proposed by Li et al. [1] is modified by introducing re-infections into the susceptible human compartment. The method of next generation matrix is used to calculate the reproduction number. We also establish the local and global stability of the equilibria using Lyapunov functions. Finally, we use numerical simulations to validate our results.

OMS-Vol. 3 (2019), Issue 1, pp. 404 – 416 Open Access Full-Text PDF

Abstract: Let \(G\) be a simple graph with vertex set \(V(G)\) and edge set \(E(G)\). A mapping \(g:V (G)\rightarrow\{1,2,…t\}\) is called \(t\)-coloring if for every edge \(e = (u, v)\), we have \(g(u) \neq g(v)\). The chromatic number of the graph \(G\) is the minimum number of colors that are required to properly color the graph. The chromatic polynomial of the graph \(G\), denoted by \(P(G, t)\) is the number of all possible proper coloring of \(G\). Dendrimers are hyper-branched macromolecules, with a rigorously tailored architecture. They can be synthesized in a controlled manner either by a divergent or a convergent procedure. Dendrimers have gained a wide range of applications in supra-molecular chemistry, particularly in host guest reactions and self-assembly processes. Their applications in chemistry, biology and nano-science are unlimited. In this paper, the chromatic polynomials for certain families of dendrimer nanostars have been computed.

OMS-Vol. 3 (2019), Issue 1, pp. 398 – 403 Open Access Full-Text PDF

Abstract: In this paper, Petrović’s inequality is generalized for \(h-\)convex functions on coordinates with the condition that \(h\) is supermultiplicative. In the case, when \(h\) is submultiplicative, Petrović’s inequality is generalized for \(h-\)concave functions. Also particular cases for \(P-\)function, Godunova-Levin functions, \(s-\)Godunova-Levin functions and \(s-\)convex functions has been discussed.

OMS-Vol. 3 (2019), Issue 1, pp. 390 – 397 Open Access Full-Text PDF

Abstract: Inspired by the observation that adjacent vertices need possess their own characteristics in terms of total coloring, we study the smarandachely adjacent vertex total coloring (abbreviated as SAVTC) of a graph \(G\), which is a proper total coloring of \(G\) such that for every vertex \(u\) and its every neighbor \(v\), the color-set of \(u\) contains a color not in the color-set of \(v\), where the color-set of a vertex is the set of colors appearing at the vertex or its incident edges. The minimum number of colors required for an SAVTC is denoted by \(\chi_{sat}(G)\). Compared with total coloring, SAVTC would be more likely to be developed for potential applications in practice. For any graph \(G\), it is clear that \(\chi_{sat}(G)\geq \Delta(G)+2\), where \(\Delta(G)\) is the maximum degree of \(G\). We, in this work, analyze this parameter for general subcubic graphs. We prove that \(\chi_{sat}(G)\leq 6\) for every subcubic graph \(G\). Especially, if \(G\) is an outerplanar or claw-free subcubic graph, then \(\chi_{sat}(G)=5\).

OMS-Vol. 3 (2019), Issue 1, pp. 382 – 389 Open Access Full-Text PDF

Abstract: In this paper, new sufficient conditions are obtained for oscillation of second-order neutral delay differential equations of the form \(\frac{d}{dt} \Biggl[r(t) \frac{d}{dt} \biggl [x(t)+p(t)x(t-\tau)\biggr]\Biggr]+q(t)G\bigl(x(t-\sigma_1)\bigr)+v(t)H\bigl(x(t-\sigma_2)\bigr)=0, \;\; t \geq t_0,\) under the assumptions \(\int_{0}^{\infty}\frac{d\eta}{r(\eta)}=\infty\) and \(\int_{0}^{\infty}\frac{d\eta}{r(\eta)}<\infty\) for \(|p(t)|<+\infty\). Two illustrative examples are included.