Spectral gap of the largest eigenvalue of the normalized graph Laplacian

TL;DR

This paper introduces a new method to establish bounds on the largest eigenvalue of the normalized graph Laplacian, revealing conditions for equality and relating it to graph complement structures.

Contribution

The paper presents a novel proof technique for eigenvalue bounds and characterizes extremal graphs where equality holds.

Findings

Max eigenvalue ≥ (n+1)/(n-1) for non-complete graphs

Equality occurs iff complement is a single edge or a complete bipartite graph

Provides a lower bound based on minimum vertex degree

Abstract

We offer a new method for proving that the maximal eigenvalue of the normalized graph Laplacian of a graph with $n$ vertices is at least $\frac{n+1}{n-1}$ provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size $\frac{n-1}2$. With the same method, we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree, provided this is at most $\frac{n-1}{2}$.