Iranian Journal of Mathematical Chemistry
Volume:12 Issue: 1, Winter 2021

Recently, a bond-additive topological descriptor, named as the Mostar index, has been introduced as a measure of peripherality in networks. For a connected graph $G$, the Mostar index is defined as $Mo(G) = sum_{e=uv in E(G)} |n_u(e) - n_v(e)|$, where for an edge $e=uv$ we denote by $n_u(e)$ the number of vertices of $G$ that are closer to $u$ than to $v$ and by $n_v(e)$ the number of vertices of $G$ that are closer to $v$ than to $u$. In the present paper, we prove that the Mostar index of a weighted graph can be computed in terms of Mostar indices of weighted quotient graphs. Inspired by this result, several generalizations to other versions of the Mostar index already appeared in the literature. Furthermore, we apply the obtained method to benzenoid systems, tree-like polyphenyl systems, and to a fullerene patch. Closed-form formulas for two families of these molecular graphs are also deduced.

In [13] Gutman introduced a novel graph invariant called Sombor index SO, defined via $sqrt{deg(u)^{2}+deg(v)^{2}}.$ In this paper we provide relations between Sombor index and some degree-based topological indices: Zagreb indices, Forgotten index and Randi' {c} index. Similar relations are established in the class of triangle-free graphs.

Let $G=(V,E)$ be a finite simple graph. The Sombor index $SO(G)$ of $G$ is defined as $sum_{uvin E(G)}sqrt{d_u^2+d_v^2}$, where $d_u$ is the degree of vertex $u$ in $G$. In this paper, we study this index for certain graphs and we examine the effects on $SO(G)$ when $G$ is modified by operations on vertex and edge of $G$. Also we present bounds for the Sombor index of join and corona product of two graphs.

In this paper, we consider a stochastic Selkov model for the glycolysis process. The stochasticity in the model is introduced by parameter perturbation which is a standard technique in stochastic mathematical modeling. First, we employ polar coordinate transformation and stochastic averaging method to transform the original system into an Itô averaging diffusion system. Next, we investigate the stochastic dynamical bifurcation of the Itô averaging amplitude equation by studying the qualitative changes of invariant measures and explore the phenomenological bifurcation (P-bifurcation) by using the counterpart Fokker-Planck equation. Finally, some numerical simulations are presented to verify our analytic arguments.

In this paper‎, ‎the chaotic properties of‎ ‎the following Belusov-Zhabotinskii's reaction model is explored:‎ ‎alk+1=(1-η)θ(‎alk)+(1/2) η[θ(‎al-1k)-θ(al+1k)], where k is discrete‎ ‎time index‎, ‎l is lattice side index with system size M‎, η∊ ‎[0‎, ‎1) is coupling constant and $theta$ is a continuous map on‎ ‎W=[-1‎, ‎1]. This kind of system is a generalization of the chemical‎ ‎reaction model which was presented by García Guirao and Lampart‎ ‎in [Chaos of a coupled lattice system related with the Belusov–Zhabotinskii reaction, J. Math. Chem. ‎48 (2010) 159-164] and stated by Kaneko in [Globally coupled chaos violates the law of large numbers but not the central-limit theorem, Phys. Rev.‎ ‎Lett‎. ‎65‎ (1990) ‎1391-1394]‎, ‎and it is closely related to the‎ ‎Belusov-Zhabotinskii's reaction‎. ‎In particular‎, ‎it is shown that for‎ ‎any coupling constant η ∊ [0‎, ‎1/2)‎, ‎any‎ ‎r ∊ {1‎, ‎2‎, ...} and θ=Qr‎, ‎the topological entropy‎ ‎of this system is greater than or equal to rlog(2-2η)‎, ‎and‎ ‎that this system is Li-Yorke chaotic and distributionally chaotic,‎ ‎where the map Q is defined by‎ ‎Q(a)=1-|1-2a|‎, ‎ a ∊ [0‎, ‎1], and Q(a)=-Q(-a),‎ a ∊ [-1‎, ‎0]. Moreover‎, ‎we also show that for any c‎, ‎d with‎ ‎0≤c≤ d≤ 1, ‎η=0 and θ=Q‎, ‎this system is‎ ‎distributionally (c‎, ‎d)-chaotic.‎