Journal of Discrete Mathematics and Its Applications
Volume:8 Issue: 1, Winter 2023

Ali Reza was my friend. I firmly believe that this friendship was mutual. In my long academic career, I had scientific cooperation with numerous colleagues (more than 800), but very few of them where my friends. Somewhere in the first years of the present century, I noticed that in Iran there is a mathematician producing remarkable results in Mathematical Chemistry. Soon we got in touch, soon we started to do joint research, soon Ali Reza became member of Editorial Board of our “MATCH Communications in Mathematical and in Computer Chemistry” (where he published a total of 43 contributions). Eventually, we met on many mathematical chemistry conferences held in Iran, when I experienced the unforgettableIranian hospitality. We also met on a couple of conferences in Dubrovnik, Croatia, on one of which Ali Reza was elected member of the International Academy of Mathematical Chemistry (to what I contributed a bit).

Ali Reza Ashrafi (10 May 1964 9 January 2023) was an Iranian mathematician who worked in computational group theory and mathematical chemistry. Ashrafi was a professor at the Department of Pure Mathematics of the University of Kashan.

In Chemical Graph Theory, several degree-based topological indices were introduced and studied since 1947. Some of these have important applications in chemistry while some only have nice mathematical properties. In this paper, we introduce the Nirmala leap index and modified Nirmala leap index, and their polynomials and compute these indices for some important nanostructures which appeared in nanoscience, especially used as a treatment alternative against Covid-19 in many countries.

The Hosoya index $Z(G)$ of a graph $G$ is the total number of matchings in it. In this paper, the recursive formulas of the Hosoya index of semitotal graph $Q(G)$ and total graph $T(G)$ for certain graphs $G$ are obtained. Moreover, we obtain the bounds of the Hosoya index of semitotal and total graphs of a connected graph $G$.

We present a new and simple direct approach based on generalized moving least squares (GMLS) for computing the first derivatives of the functions defined on the sphere. The novel method utilizes a Householder transformation (reflection) and a projection onto the tangent plane to compute the first derivatives at the original point on the sphere. The main benefit of this algorithm is that there is no need to use the spherical harmonics for constructing the approximation of the first derivatives. An example of the approximation has been tested to show the ability of the developed method. Moreover, this method has been applied to solve the transport equation in one example.

Let G be a group and R,L,S be subsets of G such that $R=R^{-1}$, $L=L^{-1}$ and $1\notin R\cup L$. The undirected graph $\SC(G;R,L,S)$ with vertex set  union of $G_1=\{g_1\mid g\in G\}$ and $G_2=\{g_2\mid g\in G\}$, and edge set the union of $\{\{g_1,(gr)_1\}\mid g\in G, r\in R\}$, $\{\{g_2,(gl)_2\}\mid g\in G,l\in L\}$ and $\{\{g_1,(gs)_2\}\mid g\in G,s\in S\}$ is called semi-Cayley graph over G.  We say that $\SC(G;R,L,S)$ is quasi-abelian if R,L and S are a  union of conjugacy classes of G. In this paper, we study the automorphism group of quasi-abelian semi-Cayley graphs.