Spectral moments of trees with given degree sequence

TL;DR

This paper proves that among trees with a fixed degree sequence, the greedy tree maximizes all spectral moments, leading to solutions for conjectures related to Estrada index and spectral properties.

Contribution

It establishes that the greedy tree maximizes spectral moments for trees with a given degree sequence, confirming related conjectures and advancing spectral graph theory.

Findings

Greedy tree maximizes spectral moments for given degree sequences.

Confirms conjecture on trees with fixed maximum degree.

Implications for Estrada index of such trees.

Abstract

Let $\lambda_1,\dots,\lambda_n$ be the eigenvalues of a graph $G$. For any $k\geq 0$, the $k$-th spectral moment of $G$ is defined by $\M_k(G)=\lambda_1^k+\dots+\lambda_n^k$. We use the fact that $\M_k(G)$ is also the number of closed walks of length $k$ in $G$ to show that among trees $T$ whose degree sequence is $D$ or majorized by $D$, $\M_k(T)$ is maximized by the greedy tree with degree sequence $D$ (constructed by assigning the highest degree in $D$ to the root, the second-, third-, \dots highest degrees to the neighbors of the root, and so on) for any $k\geq 0$. Several corollaries follow, in particular a conjecture of Ili\'c and Stevanovi\'c on trees with given maximum degree, which in turn implies a conjecture of Gutman, Furtula, Markovi\'c and Gli\v{s}i\'c on the Estrada index of such trees, which is defined as $\EE(G)=e^{\lambda_1}+\dots+e^{\lambda_n}$.