Majorization and the spectral radius of starlike trees

Abstract

A starlike tree is a tree with exactly one vertex of degree greater than two. The spectral radius of a graph G, that is denoted by $\lambda (G)$, is the largest eigenvalue of G. Let k and $n_1,\ldots ,n_k$ be some positive integers. Let $T(n_1,\ldots ,n_k)$ be the tree T (T is a path or a starlike tree) such that T has a vertex v so that $T{\setminus } v$ is the disjoint union of the paths $P_{n_1-1},\ldots ,P_{n_k-1}$ where every neighbor of v in T has degree one or two. Let $P=(p_1,\ldots ,p_k)$ and $Q=(q_1,\ldots ,q_k)$, where $p_1\ge \cdots \ge p_k\ge 1$ and $q_1\ge \cdots \ge q_k\ge 1$ are integer. We say P majorizes Q and let $P\succeq _M Q$, if for every j, $1\le j\le k$, $\sum _{i=1}^{j}p_i\ge \sum _{i=1}^{j}q_i$, with equality if $j=k$. In this paper we show that if P majorizes Q, that is $(p_1,\ldots ,p_k)\succeq _M(q_1,\ldots ,q_k)$, then $\lambda (T(q_1,\ldots ,q_k))\ge \lambda (T(p_1,\ldots ,p_k))$.