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