On Spectral Properties of Lanzhou Matrix of Graphs

TL;DR

This paper introduces the Lanzhou matrix of graphs, explores its spectral properties, and provides bounds, eigenvalues, and characterizations for specific graph classes, advancing spectral graph theory understanding.

Contribution

The paper defines the Lanzhou matrix, studies its spectral properties, and derives bounds and characterizations for eigenvalues and inertia for various graphs.

Findings

Bounds on Lanzhou energy and spread established

Lanzhou eigenvalues and inertia computed for standard graphs

Conditions for symmetry of Lanzhou eigenvalues about the origin identified

Abstract

Let $\Gamma$ be a simple graph on $n$ vertices. Lanzhou index is defined as $Lz(\Gamma)=\sum\limits_{u \in V(\Gamma)}d_\Gamma(u)^2d_{\overline{\Gamma}}(u).$ In this manuscript, the Lanzhou matrix, denoted by $A_{Lz}(\Gamma)$, has been defined, and its spectral properties are studied. The $uv^{th}$ entry in $A_{Lz}(\Gamma)$ is $d_\Gamma(u)d_{\overline{\Gamma}}(u)+d_\Gamma(v)d_{\overline{\Gamma}}(v)$ if $u$ and $v$ are adjacent. Otherwise, the entry is zero. Some bounds on Lanzhou energy and spread on the Lanzhou matrix are obtained. Also, Lanzhou eigenvalues and inertia for some standard graphs have been obtained. Additionally, characterizations for the symmetricity of Lanzhou eigenvalues about the origin are obtained.