Majorization and the spectral radius of starlike trees

Majorization and the spectral radius of starlike trees 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 $$\lambda (G)$$ λ ( G ) , is the largest eigenvalue of G. Let k and $$n_1,\ldots ,n_k$$ n 1 , … , n k be some positive integers. Let $$T(n_1,\ldots ,n_k)$$ T ( n 1 , … , n k ) be the tree T (T is a path or a starlike tree) such that T has a vertex v so that $$T{\setminus } v$$ T \ v is the disjoint union of the paths $$P_{n_1-1},\ldots ,P_{n_k-1}$$ P n 1 - 1 , … , P n k - 1 where every neighbor of v in T has degree one or two. Let $$P=(p_1,\ldots ,p_k)$$ P = ( p 1 , … , p k ) and $$Q=(q_1,\ldots ,q_k)$$ Q = ( q 1 , … , q k ) , where $$p_1\ge \cdots \ge p_k\ge 1$$ p 1 ≥ ⋯ ≥ p k ≥ 1 and $$q_1\ge \cdots \ge q_k\ge 1$$ q 1 ≥ ⋯ ≥ q k ≥ 1 are integer. We say P majorizes Q and let $$P\succeq _M Q$$ P ⪰ M Q , if for every j, $$1\le j\le k$$ 1 ≤ j ≤ k , $$\sum _{i=1}^{j}p_i\ge \sum _{i=1}^{j}q_i$$ ∑ i = 1 j p i ≥ ∑ i = 1 j q i , with equality if $$j=k$$ j = k . In this paper we show that if P majorizes Q, that is $$(p_1,\ldots ,p_k)\succeq _M(q_1,\ldots ,q_k)$$ ( p 1 , … , p k ) ⪰ M ( q 1 , … , q k ) , then $$\lambda (T(q_1,\ldots ,q_k))\ge \lambda (T(p_1,\ldots ,p_k))$$ λ ( T ( q 1 , … , q k ) ) ≥ λ ( T ( p 1 , … , p k ) ) . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Combinatorial Optimization Springer Journals

Majorization and the spectral radius of starlike trees

, Volume 36 (1) – Apr 13, 2018
9 pages

Publisher
Springer Journals
Copyright © 2018 by Springer Science+Business Media, LLC, part of Springer Nature
Subject
Mathematics; Combinatorics; Convex and Discrete Geometry; Mathematical Modeling and Industrial Mathematics; Theory of Computation; Optimization; Operations Research/Decision Theory
ISSN
1382-6905
eISSN
1573-2886
D.O.I.
10.1007/s10878-018-0287-5
Publisher site
See Article on Publisher Site

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

Journal

Journal of Combinatorial OptimizationSpringer Journals

Published: Apr 13, 2018

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