Convex combination of first and second eigenvalues of trees

TL;DR

This paper studies the spectral sum of trees and identifies extremal trees that maximize or minimize the sum and convex combinations of the largest and second largest eigenvalues, revealing new extremal properties.

Contribution

It determines extremal trees for the spectral sum and convex combinations of eigenvalues in the class of n-vertex trees, extending spectral graph theory results.

Findings

Identified trees that maximize/minimize the spectral sum of eigenvalues.

Characterized extremal trees for convex combinations of the first two eigenvalues.

Provided a comprehensive analysis of eigenvalue extremal problems in trees.

Abstract

For a graph $G$, let $\lambda_1(G)$ and $\lambda_2(G)$ denote the largest and the second largest adjacency eigenvalue of $G$. The sum $\lambda_1(G) + \lambda_2(G)$ is called the \emph{spectral sum} of $G$. We investigate the spectral sum of trees of order $n$ and determine the extremal trees that achieve maximum/minimum. Moreover, for any $\alpha \in [0,1]$, we determine the extremal trees which maximize the convex combination $\alpha \lambda_1 + (1-\alpha)\lambda_2$ in the class of $n$-vertex trees.