On the spectral moments of trees with a given bipartition

TL;DR

This paper characterizes the last four trees in spectral moment order within a specific class of bipartite trees, based on their spectral moments derived from adjacency matrix eigenvalues.

Contribution

It identifies the extremal trees in spectral moment order among bipartite trees with fixed bipartition sizes, extending understanding of spectral properties in tree structures.

Findings

Identifies the last four trees in spectral order within the class

Provides explicit characterization of these extremal trees

Enhances understanding of spectral properties in bipartite trees

Abstract

For two given positive integers $p$ and $q$ with $p\leqslant q$, we denote $\mathscr{T}_n^{p, q}={T: T$ is a tree of order $n$ with a $(p, q)$-bipartition}. For a graph $G$ with $n$ vertices, let $A(G)$ be its adjacency matrix with eigenvalues $\lambda_1(G), \lambda_2(G), ..., \lambda_n(G)$ in non-increasing order. The number $S_k(G):=\sum_{i=1}^{n}\lambda_i^k(G)\,(k=0, 1, ..., n-1)$ is called the $k$th spectral moment of $G$. Let $S(G)=(S_0(G), S_1(G),..., S_{n-1}(G))$ be the sequence of spectral moments of $G$. For two graphs $G_1$ and $G_2$, one has $G_1\prec_s G_2$ if for some $k\in {1,2,...,n-1}$, $S_i(G_1)=S_i(G_2) (i=0,1,...,k-1)$ and $S_k(G_1)<S_k(G_2)$ holds. In this paper, the last four trees, in the $S$-order, among $\mathscr{T}_n^{p, q} (4\leqslant p\leqslant q)$ are characterized.