On Spectral Properties of Lanzhou Matrix of Graphs

TL;DR

The paper introduces the Lanzhou matrix $A_{Lz}(\,\Gamma\,)$, defining its spectrum, energy, and inertia, and explores its spectral properties within graph theory. It develops fundamental identities, bounds for spread and energy, and exact spectra for several standard graphs, while establishing inertia results for paths and a symmetry criterion for the Lanzhou eigenvalues. The work reveals that $A_{Lz}(\,\Gamma\,)$ can exhibit substantially different spectral behavior from the adjacency matrix, including energy/structure relationships under regularity and graph operations. These results provide a novel degree-complementary spectral descriptor with potential applications in graph learning and chemical graph theory.

Abstract

Let $Γ$ be a simple graph on $n$ vertices. Lanzhou index is defined as $Lz(Γ)=\sum\limits_{u \in V(Γ)}d_Γ(u)^2d_{\overlineΓ}(u).$ In this manuscript, the Lanzhou matrix, denoted by $A_{Lz}(Γ)$, has been defined, and its spectral properties are studied. The $uv^{th}$ entry in $A_{Lz}(Γ)$ is $d_Γ(u)d_{\overlineΓ}(u)+d_Γ(v)d_{\overlineΓ}(v)$ 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.

Theorems & Definitions (65)

• Definition 1
• Definition 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7
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• Lemma 9
• Lemma 10