Diminished Sombor matrix, spectral radius, and energy of the graphs

TL;DR

This paper introduces the diminished Sombor matrix for graphs, establishes bounds for its spectral radius and energy, and characterizes the extremal graphs that attain these bounds.

Contribution

It defines a new matrix associated with graphs and derives sharp bounds for its spectral properties, identifying extremal graphs.

Findings

Sharp bounds for spectral radius of the diminished Sombor matrix

Sharp bounds for energy of the diminished Sombor matrix

Characterization of graphs attaining extremal bounds

Abstract

Consider a simple graph $G$ with vertex set $V = \{v_1, v_2, \ldots, v_n\}$ and edge set $E$. The diminished Sombor matrix $M_{DS}(G)$ is constructed such that its $(i, j)$ entry is $\frac{\sqrt{d_i^2+d_j^2}}{d_i+d_j}$ if vertices $v_iv_j \in E$, and $0$ otherwise, where $d_i$ represents the degree of vertex $v_i$. In this paper, we establish sharp bounds for the spectral radius, and energy of the Sombor matrix of graphs and identify the graphs that attain these extremal values.