On the spectral moments of trees with a given bipartition

Abstract

For two given positive integers $p$ and $q$ with $p⩽q$, 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 ${\mathcal{T}}_n^{p,q}(4⩽p⩽q)$ are characterized.

11 pages, 7 figures

References in corpus (3)

• On the spectral moment of trees with given degree sequences
• On the spectral moment of graphs with $k$ cut edges
• On the spectral moment of graphs with given clique number