ScholarWorks@성균관대학교

초록

Let G be a simple graph with vertex set V (G) = {v(1), v(2), ... , v(n)} and edge set E(G), where |V (G)|= n and |E(G)|= m. Molecular descriptors play a significant role in quantitative studies of structure-property and structure-activity relationships. One of the popular degree-based topological indices, the Sombor index (SO), is a chemically useful descriptor. The Sombor index of a graph G is defined as SO(G) = Sigma(vivj is an element of E(G)) root d(i)(2) + d(j)(2), where d(i )is the degree of the vertex v(i )is an element of V(G). The Sombor matrix of G, denoted by SM(G), is defined as the n x n matrix whose (i, j)-entry is root d(i)(2) + d(j)(2) if v(i)v(j) is an element of E(G), and 0 otherwise. Let the eigenvalues of the Sombor matrix SM(G) be sigma(1) >= sigma(2)>= <middle dot><middle dot> <middle dot>>= sigma(n). Very recently, Rabizadeh, Habibi and Gutman [Some notes on Sombor index of graphs, MATCH Commun. Math. Comput. Chem. 93 (2025) 853-859] proposed a conjecture about the Sombor index of graphs, stated as follows: (a) m|sigma(n)| =-m sigma(n) >= SO(G). (b) If G is connected, then the equality in (a) holds if and only if G is a complete graph. In the general case, equality holds if and only if G consists of mutually isomorphic complete graphs and some (or no) isolated vertices. In this paper, we provide a complete solution to the above conjecture.

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