Convex combination of first and second eigenvalues of trees

TL;DR

This paper analyzes the spectral sum Ψ(T,α)=αλ1(T)+(1−α)λ2(T) of trees and identifies the extremal structures in the n-vertex tree class. It shows that for 0≤α≤1/2 the extremals are balanced double comets DC(k1,k2,ℓ), with explicit forms depending on parity and α, while for α>1/2 the maximizers become two-ended double comets DC(tn+C,(1−t)n−C−2,2) with t=α^2/(α^2+(1−α)^2). The authors also establish the asymptotic behavior of the normalized maximum and prove that the minimum spectral sum over trees is attained by the path P_n for large n (with stars dominating only for very small n). The results unify spectral-extremal phenomena for λ1 and λ2 on trees, provide explicit constructions (double comets) as extremals, and open avenues for extending these ideas to broader graph families and related spectral measures.

Abstract

For a graph $G$, let $λ_1(G)$ and $λ_2(G)$ denote the largest and the second largest adjacency eigenvalue of $G$. The sum $λ_1(G) + λ_2(G)$ 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 $α\in [0,1]$, we determine the extremal trees which maximize the convex combination $αλ_1 + (1-α)λ_2$ in the class of $n$-vertex trees.

Figures (4)

• Figure 1: $DC(k_1, k_2, \ell)$
• Figure 2: The black curve represents $\hat{\Psi}_6(\alpha)$. The different colored lines represent $\frac{\Psi(T, \alpha)}{\sqrt{5}}$ for different trees $T$ on 6 vertices.
• Figure 3: Limit of $\hat{\Psi}_n(\alpha)$
• Figure 4: $T^{\#}$ and $T'$

Theorems & Definitions (76)

• Theorem 1.1: Lovasz_Pelikan_1973
• Theorem 1.2: Neumaier_1982Hofmeister_1997
• Conjecture 1.3: Stanic_2018
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Lemma 2.1: CRS_2010
• Proposition 2.2