Journal of Combinatorial Optimization

, Volume 36, Issue 1, pp 121–129

# Majorization and the spectral radius of starlike trees

Article

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

## Mathematics Subject Classification

05C31 05C50 15A18

## Notes

### Acknowledgements

This research was in part supported by a grant (No. 96050011) from School of Mathematics, Institute for Research in Fundamental Sciences (IPM).

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