On graphs with three or four distinct normalized Laplacian eigenvalues

TL;DR

This paper characterizes specific classes of graphs based on their normalized Laplacian eigenvalues, focusing on graphs with three or four distinct eigenvalues and their structural properties.

Contribution

It provides a complete characterization of connected graphs with three or four normalized Laplacian eigenvalues, including bipartite and unicyclic graphs, with particular eigenvalue conditions.

Findings

Characterized all connected graphs with exactly three normalized Laplacian eigenvalues, one of which is 1.

Determined all bipartite graphs with at least one degree-1 vertex having four eigenvalues.

Identified all unicyclic graphs with three or four normalized Laplacian eigenvalues.

Abstract

In this paper, we characterize all connected graphs with exactly three distinct normalized Laplacian eigenvalues of which one is equal to $1$, determine all connected bipartite graphs with at least one vertex of degree $1$ having exactly four distinct normalized Laplacian eigenvalues, and find all unicyclic graphs with three or four distinct normalized Laplacian eigenvalues.