Journal of Discrete Mathematics and Its Applications
Volume:8 Issue: 3, Summer 2023

The field of mathematical chemistry started with the pure and memorable activities of Alireza Ashrafi in Iran. One of the most important factors in the rapid expansion of this branchof science in Iran, which intensified almost since 2003, has been the regular holding of mathematical chemistry conferences in Iran which were Iranian version of MCC conferences in Europe. One of the prominent features of Ashrafi was his active and effective presence in international scientific communication and diplomacy, and at the same time helding international conference on mathematical chemistry in Iran. With his invitation, many scientists from all over the world came to Iran and scientific diplomacy in his field of expertise became popular in the real sense. The mathematical chemistry conferences in Iran were a series of academic conferences that focus on the intersection of mathematics and chemistry. These conferences have been held annually since 2005 and were organized by prominent Iranian mathematicians and chemists, including Alireza Ashrafi and Ali Iranmanesh. The conferences brought together researchers, professors, and students from Iran and around the world to discuss topics such as chemical graphs, topological indices, and mathematical modeling of chemicalsystems. These conferences have helped to promote the development of mathematical chemistryin Iran and have facilitated collaborations between Iranian and international researchers. The conferences have also produced numerous publications in top scientific journals and have helped to establish Iran as a leading country in the field of mathematical chemistry.

The symmetric division deg index (or simply SDD) was proposed by Vukicevic et al.as a remarkable predictor of total surface area of polychlorobiphenyls. We are interested in how the SDD of a graph changes when edges are deleted. The obtained results show that all cases are possible: increased, decreased and unchanged. In this article, we present some necessary conditions for the occurrence of each of the three different states.

Let $G$ be a simple graph with order $n$ and size $m$. Let $D(G)=$ diag$(d_1, d_2, \dots, d_n)$ be its diagonal matrix, where $d_i=\deg(v_i),$ for all $i=1,2,\dots,n$ and $A(G)$ be its adjacency matrix. The matrix $Q(G)=D(G)+A(G)$ is called the signless Laplacian matrix of $G$. Let $q_1,q_2,\dots,q_n$ be the signless Laplacian eigenvalues of $Q(G)$ and let $S^{+}_{k}(G)=\sum_{i=1}^{k}q_i$ be the sum of the $k$ largest signless Laplacian eigenvalues. Ashraf et al. [F. Ashraf, G. R. Omidi, B. Tayfeh-Rezaie, On the sum of signless Laplacian eigenvalues of a graph, Linear Algebra Appl. {\bf 438} (2013) 4539-4546.] conjectured that $S^{+}_{k}(G)\leq m+{k+1 \choose 2}$, for all $k=1,2,\dots,n$. We present a survey about the developments of this conjecture.

‎The adjacency matrix is important invariant of a graph with a chemical meaning‎, ‎when we study the chemical graphs‎. ‎In this paper‎, ‎the general form of the adjacency matrices of some hexagonal systems will be determined‎.

Let $\Gamma$ be a $k-$regular graph with the second maximum  eigenvalue $\lambda$. Then  $\Gamma$ is said o be Ramanujan graph if $\lambda\leq 2\sqrt{k-1}.$ Let $G$ be a finite group  and $\Gamma=Cay(G,S)$ be a Cayley graph related to $G$.  The aim of this paper is to investigate the Ramanujan Cayley graphs of sporadic groups.

A power graph is defined a graph that it’s vertices are the elements of group and two vertices are adjacent if and only if one of them is a power of the other. Suppose A(X) is the adjacency matrix of graph X. Then the polynomial χ(X,λ) = det(xI − A(X)) is called as characteristic polynomial of X. In this paper, we compute the characteristic polynomial of all power graphs of order p2q, where p, q are distinct prime numbers.