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Mathematical Problems in Engineering
Volume 2016, Article ID 4702387, 7 pages
http://dx.doi.org/10.1155/2016/4702387
Research Article

Maximum Matchings of a Digraph Based on the Largest Geometric Multiplicity

College of Information Engineering, Taiyuan University of Technology, Taiyuan, Shanxi 030024, China

Received 6 December 2015; Accepted 10 April 2016

Academic Editor: Zhike Peng

Copyright © 2016 Yunyun Yang and Gang Xie. 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

Matching theory is one of the most forefront issues of graph theory. Based on the largest geometric multiplicity, we develop an efficient approach to identify maximum matchings in a digraph. For a given digraph, it has been proved that the number of maximum matched nodes has close relationship with the largest geometric multiplicity of the transpose of the adjacency matrix. Moreover, through fundamental column transformations, we can obtain the matched nodes and related matching edges. In particular, when a digraph contains a cycle factor, the largest geometric multiplicity is equal to one. In this case, the maximum matching is a perfect matching and each node in the digraph is a matched node. The method is validated by an example.