Laplacian integral graphs with a given degree sequence constraint

TL;DR

This paper characterizes all connected non-bipartite graphs with at most two vertices of degree ≥ 3 that have integer Laplacian eigenvalues, contributing to the understanding of L-integral graphs under degree constraints.

Contribution

It provides a complete characterization of L-integral non-bipartite graphs with limited high-degree vertices among connected graphs.

Findings

Identifies all such L-integral graphs under the given degree constraints.

Provides a classification based on the degree sequence and connectivity.

Enhances understanding of spectral properties of graphs with degree restrictions.

Abstract

Let G be a graph on n vertices. The Laplacian matrix of G, denoted by L(G), is defined as L(G) = D(G) - A(G), where A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of the vertex degrees of G. A graph G is said to be L-integral is all eigenvalues of the matrix L(G) are integers. In this paper, we characterize all L-integral non-bipartite graphs among all connected graphs with at most two vertices of degree larger than or equal to three.