Iranian Journal of Mathematical Chemistry
Volume:15 Issue: 1, Winter 2024

‎The Sombor index $(SO)$ is a recently invented vertex-degree-based molecular structure-descriptor‎. ‎Let $M_1$ be the first Zagreb index‎. ‎The fact that $SO$ is bounded from below by $M_{1}/\sqrt{2}$ and from above by $M_{1}$ is well-known and easy to prove‎. ‎In this paper‎, ‎we improve these bounds‎.

‎The number of spanning trees of a graph $G$ is called the complexity of $G$‎. It is known that the complexity of the line graph of a given graph $G$ can‎ be computed as the sum over all spanning trees of $G$ of contributions‎ ‎which depend on various types of products of degrees of vertices of $G$‎. ‎We interpret the contributions in terms of three types of multiplicative‎ Zagreb indices‎, ‎obtaining simple and compact expressions for the complexity of‎ ‎line graphs of graphs with low cyclomatic numbers‎. ‎As an application‎, ‎we‎ determine the unicyclic graphs whose line graphs have the smallest and the‎ largest complexity‎.

‎The second multiplicative Zagreb eccentricity index $E^{*}_{2} ({G})$ of a simple connected‎ graph $G$ is expressed as the product of the weights‎ $\varepsilon_{G}(a)\varepsilon_{G}(b)$ over all edges $ab$ of $G$‎, where $\varepsilon_{G}(a)$ stands for the‎ eccentricity of the vertex $a$ in $G$‎. ‎In this‎ paper‎, ‎some extremal problems on the $E^{*}_{2}$ index over some special graph classes including‎ trees‎, ‎unicyclic graphs and bicyclic graphs are examined‎, ‎and‎ ‎the corresponding extremal graphs are characterized‎. ‎Besides‎, ‎the relationships between this vertex-eccentricity-based graph invariant and some well-known parameters of graphs and existing graph invariants such as the number of vertices‎, ‎number of edges‎, ‎minimum vertex degree‎, ‎maximum vertex degree‎, ‎eccentric connectivity index‎, ‎connective eccentricity index‎, ‎first multiplicative Zagreb eccentricity index and second multiplicative Zagreb index are investigated‎.

‎In this paper‎, ‎we calculate the expected values of the first and second Zagreb indices‎, ‎denoted as $\textbf{E}\left(M_1\right)$ and $\textbf{E}\left(M_2\right)$ respectively‎, ‎as well as the expected value of the forgotten index‎, ‎$\textbf{E}\left(F\right)$‎, ‎for two models of random bipartite graphs‎. ‎To evaluate our findings‎, ‎we establish the growth rate by demonstrating that for a random bipartite graph $G$ of order $n$ in either model‎, ‎the expected value of $M_1(G)$ is $O\left( n^3 \right)$‎. ‎Furthermore‎, ‎we prove that the expected values of $M_2(G)$ and $F(G)$ are both $O\left( n^4 \right)$‎.

‎Let $ \sum_{i=0}^{n}(-1)^il_ix^{n-i}$ and $\sum_{i=0}^{n}(-1)^iq_ix^{n-i}$ be the characteristic polynomials of the Laplacian matrix and signless Laplacian matrix of an $n$-vertex graph‎, ‎respectively‎. ‎Let $\alpha_i = |q_i‎ - ‎l_i|$‎, ‎$0\leq i \leq n$‎. ‎In this paper‎, ‎we find formulas for some of $\alpha_i$'s‎. ‎In particular‎, ‎we compute $\alpha_i$'s for some fullerene graphs.