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Discrete Dynamics in Nature and Society
Volume 2017 (2017), Article ID 1382980, 8 pages
https://doi.org/10.1155/2017/1382980
Research Article

The Minimum Spectral Radius of an Edge-Removed Network: A Hypercube Perspective

1School of Software Engineering, Chongqing University, Chongqing 400044, China
2School of Mechanical Engineering, Chongqing University, Chongqing 400044, China

Correspondence should be addressed to Yingbo Wu; nc.ude.uqc@byw

Received 15 December 2016; Revised 17 February 2017; Accepted 28 March 2017; Published 19 April 2017

Academic Editor: Yong Deng

Copyright © 2017 Yingbo Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The spectral radius minimization problem (SRMP), which aims to minimize the spectral radius of a network by deleting a given number of edges, turns out to be crucial to containing the prevalence of an undesirable object on the network. As the SRMP is NP-hard, it is very unlikely that there is a polynomial-time algorithm for it. As a result, it is proper to focus on the development of effective and efficient heuristic algorithms for the SRMP. For that purpose, it is appropriate to gain insight into the pattern of an optimal solution to the SRMP by means of checking some regular networks. Hypercubes are a celebrated class of regular networks. This paper empirically studies the SRMP for hypercubes with two/three/four missing edges. First, for each of the three subproblems of the SRMP, a candidate for the optimal solution is presented. Second, it is shown that the candidate is optimal for small-sized hypercubes, and it is shown that the proposed candidate is likely to be optimal for medium-sized hypercubes. The edges in each candidate are evenly distributed over the network, which may be a common feature of all symmetric networks and hence is instructive in designing effective heuristic algorithms for the SRMP.