On the conjecture of Sombor energy of a graph

Abstract

The Sombor matrix of a graph G with vertices v1,v2,…,vn is defined as ASO(G)=[sij], where sij=di2+dj2 if vi is adjacent to vj and sij=0, otherwise, where di is the degree of a vertex vi. The Sombor energy of a graph is defined as the sum of the absolute values of the eigenvalues of the Sombor matrix. N. Ghanbari (Ghanbari, 2022) conjectured that there is no graph with integer valued Sombor energy. In this paper we give some class of graphs for which this conjecture holds. Further we conjecture that there is no regular graph with adjacency energy equal to 2k2 where k is a positive integer.