Abstract
<p style="margin: 0in 0in 0pt; text-align: justify;"><span style="font-size: 11pt;"><span style="font-family: Times New Roman;">A maximal matching of a graph is the set of edges such that the addition of an edge to this set violates the property of matching (i.e., no two edges of the matching share a vertex). We use the notion of assortative index (ranges from -1 to 1) to evaluate the extent of similarity of the end vertices constituting the edges of a matching. A maximal matching of the edges whose assortative index is as close as possible to 1 is referred to as maximal assortative matching (MAM) and a maximal matching of the edges whose assortative index is as close as possible to -1 is referred to as maximal dissortative matching (MDM). We present algorithms to determine the MAM and MDM of the edges in a network graph. Through extensive simulations, we conclude that random network graphs are more conducive for maximal dissortative matching rather than maximal assortative matching. We observe the assortative index of an MDM on random network graphs to be relatively more closer to the targeted optimal value of -1 compared to the assortative index of an MAM to the targeted optimal value of 1. </span></span></p>
Highlights
Maximal matching is one of the classical problems of graph theory
The results presented for random network graphs with node degree as node weights illustrate that the assortative index values obtained with Maximal Dissortative Matching (MDM) are more closer to the targeted optimal value (-1) compared to the closeness of the assortative index values obtained with the Maximal Assortative Matching (MAM) to the targeted optimal value (1)
We explored the feasibility of determining maximal matching in random network graphs with the objective of maximizing the assortative index or minimizing the assortative index
Summary
Maximal matching is one of the classical problems of graph theory. A matching for a graph is a set of edges such that no two edges share a common vertex. We emphasize that the similarity of the end vertices of the edges constituting a maximal matching needs to be taken into consideration in the design of matching algorithms for complex network graphs. Random network graphs are a category of complex network graphs for which there exists an edge between any two vertices with a probability To vindicate this characteristic, we observe the assortative index of the entire set of edges in a random network graph to be close to 0, indicating that there is no particular preference for a node to have an edge to any other node.
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