On the Sombor characteristic polynomial and Sombor energy of a graph

Abstract

Let 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.

Strong domatic number of a graph

“…As a consequence, they have shown that the Petersen graph is determined uniquely by its domination polynomial. Ghanbari has studied the Sombor characteristic polynomial and Sombor energy of these graphs in [7], and has shown that the Petersen graph is not determined uniquely by its Sombor energy, but it has the maximum Sombor energy among others.…”

On Maximum Degree and Maximum Reverse Degree Energies of Splitting and Shadow graph of Complete graph

“…Statistical information on the research of graph energies and their applications is provided in [13]. [14] introduces the Sombor characteristic polynomial and the Sombor energy for some graph classes. In [15], the author establishes the relationship between energy and Sombor energy of the m-splitting graph and m-shadow graph of a k-regular graph.…”