The unicyclic graphs with the second smallest normalized Laplacian eigenvalue no less than $1-\frac{\sqrt{6}}{3}$

TL;DR

This paper characterizes all unicyclic graphs of order at least 21 with a second smallest normalized Laplacian eigenvalue above a specific threshold, and precisely identifies those achieving equality.

Contribution

It provides a complete classification of unicyclic graphs with second normalized Laplacian eigenvalue at least $1-rac{ ext{sqrt}(6)}{3}$, including the exact graphs attaining this bound.

Findings

Identifies all unicyclic graphs with eigenvalue ≥ $1-rac{ ext{sqrt}(6)}{3}$ for n ≥ 21.

Determines the exact unicyclic graphs where the eigenvalue equals $1-rac{ ext{sqrt}(6)}{3}$.

Abstract

Let $\lambda_{2}(G)$ be the second smallest normalized Laplacian eigenvalue of a graph $G$. In this paper, we determine all unicyclic graphs of order $n\geq21$ with $\lambda_{2}(G)\geq 1-\frac{\sqrt{6}}{3}$. Moreover, the unicyclic graphs with $\lambda_{2}(G)=1-\frac{\sqrt{6}}{3}$ are also determined.