Classification of graphs by Laplacian eigenvalue distribution and independence number

TL;DR

This paper characterizes specific classes of graphs based on the distribution of Laplacian eigenvalues and their independence number, identifying unique structures and spectral properties for certain parameters.

Contribution

It determines classes of graphs satisfying a specific Laplacian eigenvalue condition for independence numbers 2 and n-2, including binary star graphs.

Findings

Graphs with independence number 2 are isomorphic to a specific join of complete graphs.

Binary star graphs are characterized and shown to be Laplacian spectrum determined when p=r.

The paper provides explicit structural descriptions for graphs meeting the spectral condition.

Abstract

Let $m_GI$ denote the number of Laplacian eigenvalues of a graph $G$ in an interval $I$ and let $\alpha(G)$ denote the independence number of $G$. In this paper, we determine the classes of graphs that satisfy the condition $m_G[0,n-\alpha(G)]=\alpha(G)$ when $\alpha(G)= 2$ and $\alpha(G)= n-2$, where $n$ is the order of $G$. When $\alpha(G)=2$, $G \cong K_1 \nabla K_{n-m} \nabla K_{m-1}$ for some $m \geq 2$. When $\alpha(G)=n-2$, there are two types of graphs $B(p,q,r)$ and $B'(p,q,r)$ of order $n=p+q+r+2$, which we call the binary star graphs. Also, we show that the binary star graphs with $p=r$ are determined by their Laplacian spectra.