On the Sombor characteristic polynomial and Sombor energy of a graph

Abstract

AbstractLet G be a simple graph with vertex set $$V(G) = \{v_1, v_2,\ldots , v_n\}$$ V ( G ) = { v 1 , v 2 , … , v n } . The Sombor matrix of G, denoted by $$A_{SO}(G)$$ A SO ( G ) , is defined as the $$n\times n$$ n × n matrix whose (i, j)-entry is $$\sqrt{d_i^2+d_j^2}$$ d i 2 + d j 2 if $$v_i$$ v i and $$v_j$$ v j are adjacent and 0 for another cases. Let the eigenvalues of the Sombor matrix $$A_{SO}(G)$$ A SO ( G ) be $$\rho _1\ge \rho _2\ge \cdots \ge \rho _n$$ ρ 1 ≥ ρ 2 ≥ ⋯ ≥ ρ n which are the roots of the Sombor characteristic polynomial $$\prod _{i=1}^n (\rho -\rho _i)$$ ∏ i = 1 n ( ρ - ρ i ) . The Sombor energy $${E_{SO}}$$ E SO of G is the sum of absolute values of the eigenvalues of $$A_{SO}(G)$$ A SO ( G ) . In this paper, we compute the Sombor characteristic polynomial and the Sombor energy for some graph classes, define Sombor energy unique and propose a conjecture on Sombor energy.