On the second largest adjacency eigenvalue of trees with given diameter

TL;DR

This paper identifies extremal trees with maximum and minimum second largest adjacency eigenvalues within classes of trees with fixed diameter, advancing understanding of spectral extremal problems in graph theory.

Contribution

It determines the extremal trees for the second largest eigenvalue in trees with fixed diameter, providing new insights and revisiting spectral center concepts.

Findings

Identified trees with extremal $\lambda_2$ in $\mathcal{T}(n,d)$.

Established bounds for $\lambda_2$ in diameter-constrained trees.

Revisited spectral center and $\lambda_2$ maximization proofs.

Abstract

For a graph $G$, let $\lambda_2(G)$ denote the second largest eigenvalue of the adjacency matrix of $G$. We determine the extremal trees with maximum/minimum adjacency eigenvalue $\lambda_2$ in the class $\mathcal{T}(n,d)$ of $n$-vertex trees with diameter $d$. This contributes to the literature on $\lambda_2$-extremization over different graph families. We also revisit the notion of the spectral center of a tree and the proof of $\lambda_2$ maximization over trees.