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 \(\lambda (T(q_1,\ldots ,q_k))\ge \lambda (T(p_1,\ldots ,p_k))\).