Iranian Journal of Mathematical Chemistry
Volume:13 Issue: 1, Winter 2022

Among the binary operations made with graphs‎, ‎the cartesian‎, ‎corona‎, ‎and lexicographic are three well-known products‎, ‎as well as the cartesian sum‎. ‎Topological indices are graph invariants used to describe graphs associated with molecules‎, ‎one of these is the Schultz index‎, ‎which can be obtained as‎ ‎ ‎∑ u≠v (deg u+ deg v) d(u,v), ‎where the sum runs over all pairs of distinct vertices of the graph‎. ‎In this paper‎, ‎we give explicit expressions for the Schultz index of cartesian and corona‎, ‎with alternative proofs to those given in the literature‎, ‎as well as for lexicographic product and the cartesian sum‎, ‎all of these formulas involve order and size of factors‎, ‎additionally‎, ‎the first three involve both Wiener and Schultz indices of factors‎, ‎corona and lexicographic also involve Zagreb index and the last one just Zagreb.‎ ‎

Assume denotes a connected and simple graph with edge set  E(G) as well as vertex set ‎V(G). ‎In chemical graph theory‎, ‎the atom-bond connectivity index, as well as the Randić index of graph are two well-defined topological indices‎. ‎In addition‎, ‎Ali and Du [On the difference between  ABC and Randić indices of binary and chemical trees‎, ‎Int‎. ‎J‎. ‎Quantum Chem‎. ‎(2017) e25446] recently unveiled the distinction between Randić and ABC indices‎. ‎In this report‎, ‎we study the link between the difference of Randić and ABC indices with certain well-studied topological indices‎.

In QSAR and QSPR studies the most commonly used topological index was proposed by chemist Milan Randić in 1975 called Randić branching index or path-one molecular connectivity index, 1χ and it has many applications. In the extension of connectivity indices, in early 1990s, chemist Milan Randic´ introduced variable Randić index defined as ∑v1v2∈E(G) ((dv1 + θ*)(dv2 + θ*))−1/2, where θ* is a non-negative real number and dv1 is the degree of vertex V1 in G. The main objective of the present study is to prove the conjecture proposed in [19]. In this study, we will show that the Pn (path graph) has the maximum variable connectivity index among the collection of trees whose order is n, where n ≥ 4.

Let G=(V,E) be a finite simple graph. The Graovac-Ghorbani index of a graph G is defined as ABCGG(G)=∑uv∈E(G) √((nu(uv,G)+nv(uv,G)-2)/(nu(uv,G)nv(uv,G))), where nu(uv,G) is the number of vertices closer to vertex u than vertex v of the edge uv∈E(G).  nv(uv,G) is defined analogously. The atom-bond connectivity index of a graph G is defined as ABC(G)=∑uv∈E(G)√((du+dv-2)(dudv)), where du is the degree of vertex u in G. Let G be a connected graph constructed from pairwise disjoint connected graphs G1,...,Gk by selecting a vertex of G1, a vertex of G2, and identifying these two vertices. Then continue in this manner inductively. We say that G is obtained by point-attaching from G1,...,Gk and that Gi 's are the primary subgraphs of G. In this paper, we give some lower and upper bounds on Graovac-Ghorbani and atom-bond connectivity indices for these graphs. Additionally, we consider some particular cases of these graphs that are of importance in chemistry and study their Graovac-Ghorbani and atom-bond connectivity indices.

Although many indices are strongly correlated with various chemical as well as physical properties of a molecular compound but some of them lack proper geometrical interpretations. A newly introduced index called Sombor index is able to catch the attention of the researchers because of its connection with the notion of “2-norm”. In this paper, we propose the status version of the Sombor index. Further, we discuss a generalization of our proposed index and carry out QSPR analysis. Some mathematical properties of the generalization are also discussed.