Iranian Journal of Mathematical Chemistry
Volume:16 Issue: 1, Winter 2025

‎Let $G$ be a graph with vertex set $V\left(G \right)$‎. ‎The vertex-degree function index $H_{f}\left(G \right) $ is defined on $G$ as‎: ‎$H_{f}\left(G \right) =\sum_{u\in V\left(G \right)}f\left(d_{u} \right)‎,$‎where $f\left(x \right) $ is a function defined on positive real numbers‎. ‎Our main concern in this paper is to study $H_{f}$ over the set $\mathcal{T}_{n}$ of trees with $n$ vertices‎, ‎over the set $\mathcal{T}_{n,k}$ of trees with $n$ vertices and $k$ branching vertices‎, ‎and over the set $\mathcal{T}^{p}_{n}$ of trees with $n$ vertices and $p$ pendant vertices‎. ‎Namely‎, ‎we will show in each of these sets of trees that it is possible via branching operations to construct a strictly monotone sequence of trees that reaches the extremal values of $H_{f}$‎, ‎when $f\left( x+1\right)-f\left( x\right) $ is a strictly increasing function‎.

In this paper, a combined methodology based on the method of lines (MOL) and spline is implemented to simulate the solution of a two-dimensional (2D) stochastic fractional telegraph equation with Caputo fractional derivatives of order α and β where 1 < α, β ≤ 2. In this approach, the spatial directions are discretized by selecting some equidistance mesh points. Then fractional derivatives are estimated via linear spline approximation and some finite difference formulas. After substituting these estimations in the semi-discretization equation, the considered problem is transformed into a system of second-order initial value problems (IVPs), which is solved by using an ordinary differential equations (ODEs) solver technique in Matlab software. Also, it is proved that the rate of convergence is O(∆x2 + ∆y2), where ∆x and ∆y denote the spatial step size in x and y directions, respectively. Finally, two examples are included to confirm the efficiency of the suggested method.

‎Ivan Gutman has introduced two essential indices; the energy of a graph G‎, ‎and the Sombor index of that‎. ‎$\varepsilon(G)$‎, ‎which stands for the first index‎, ‎is the sum of the absolute values of all eigenvalues related to the adjacency matrix of the graph $G$‎. ‎The second‎, ‎defined as $SO(G)=\sum _{uv \in E(G)}\sqrt{d_u^2+d_v^2}$‎, ‎where $d_u$ and $d_v$ are the degrees of vertices $u$ and $v$ in $G$‎, ‎respectively‎. ‎It was proved that if $G$ is a graph of order at least 3‎, ‎then $\varepsilon(G)\leq So(G)$ and if $G$ is a connected graph of order $n$ that is not $P_n$ for $n\leq 8$‎, ‎then $\varepsilon(G)\leq \frac{So(G)}{2}$‎.‎In this paper‎, ‎we have strengthened these results and will obtain several lower and upper bounds between the energy of a graph‎, ‎Laplacian energy‎, ‎and the Sombor index‎.

‎Let $G=(V,E)$ be a simple graph‎. A function $f:V\rightarrow \mathbb{N}\cup \{0\}$ is called a configuration of pebbles on the vertices of $G$ and the quantity $\vert f\vert=\sum_{u\in V}f(u)$‎ ‎is called the weight of $f$ which is just the total number of pebbles assigned to vertices‎. ‎A pebbling step from a vertex $u$ to one of its‎ neighbors $v$ reduces $f(u)$ by two and increases $f(v)$ by one‎. ‎A pebbling configuration $f$ is said to be solvable if for every vertex $ v $‎, ‎there exists a sequence (possibly empty) of pebbling moves that results in a pebble on $v$‎. ‎The pebbling number $ \pi(G) $ equals the minimum number $ k $ such that every pebbling configuration $ f $ with $ \vert f\vert = k $ is solvable‎. Let $ G $ be a connected graph constructed from pairwise disjoint connected graphs $ G_1,...,G_k $ by selecting a vertex of $ G_1 $‎, ‎a vertex of $ G_2 $‎, ‎and identifying these two vertices‎. ‎Then continue in this manner inductively‎. ‎We say that $ G $ is a polymer graph‎, ‎obtained by point-attaching from monomer units $ G_1,...,G_k $‎. In this paper‎, ‎we study the pebbling number of some polymers‎. ‎

‎The additively weighted edge Mostar index is a topological index(TI) defined as an extension of the edge Mostar index‎. ‎In this paper‎, ‎we determine the extrema of the additively weighted edge Mostar index for trees‎. ‎Additionally‎, ‎we compute the lower bound and first four upper bounds of additively weighted edge Mostar index of unicyclic graphs and the upper bound for cacti with a fixed number of cycles‎. ‎All the graphs attaining the bounds are characterized‎. ‎We also propose two conjectures on additively weighted edge Mostar index of bicyclic graphs‎.

‎Let $G=(V(G),E(G))$ be a graph with the set of vertices $V(G)$ and the set of edges $E(G)$‎. ‎A subset $S$ of $E(G)$ is called a $k$-nearly independent edge subset if there are exactly $k$ pairs of elements of $S$ that share a common end‎. ‎$Z_k(G)$ is the number of such subsets‎.‎This paper studies $Z_1$‎. ‎Various properties of $Z_1$ are discussed‎. ‎We characterize the two $n$-vertex trees with the smallest $Z_1$‎, ‎as well as the one with the largest value‎. ‎A conjecture on the $n$-vertex tree with the second-largest $Z_1$ is proposed‎. ‎