On the Sombor characteristic polynomial and Sombor energy of a graph

TL;DR

This paper introduces the Sombor matrix and energy of a graph, computes their properties for various graph classes, and proposes a conjecture on the Sombor energy, expanding spectral graph theory.

Contribution

It defines the Sombor matrix and energy, computes their characteristics for specific graphs, and presents a new conjecture on Sombor energy.

Findings

Derived the Sombor characteristic polynomial for certain graphs.

Calculated the Sombor energy for specific graph classes.

Proposed a conjecture relating to Sombor energy.

Abstract

Let $G$ be a simple graph with vertex set $V(G) = \{v_1, v_2,\ldots, v_n\}$. The Sombor matrix of $G$, denoted by $A_{SO}(G)$, is defined as the $n\times n$ matrix whose $(i,j)$-entry is $\sqrt{d_i^2+d_j^2}$ if $v_i$ and $v_j$ are adjacent and $0$ for another cases. Let the eigenvalues of the Sombor matrix $A_{SO}(G)$ be $\rho_1\geq \rho_2\geq \ldots\geq \rho_n$ which are the roots of the Sombor characteristic polynomial $\prod_{i=1}^n (\rho-\rho_i)$. The Sombor energy $En_{SO}$ of $G$ is the sum of absolute values of the eigenvalues of $A_{SO}(G)$. 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.