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Counting Matchings and Tree-Like Walks in Regular Graphs

Published online by Cambridge University Press:  10 February 2010

IAN. M. WANLESS
IAN. M. WANLESS*
Affiliation:
School of Mathematical Sciences, Monash University, Vic 3800, Australia (e-mail: ian.wanless@sci.monash.edu.au)
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Abstract

The number of closed tree-like walks in a graph is closely related to the moments of the roots of the matching polynomial for the graph. Thus, by counting these walks up to a given length it is possible to find approximations for the matching polynomial. This approach has been used in two separate problems involving asymptotic enumerations of 1-factorizations of regular graphs. Nevertheless, a systematic way to count the required walks had not previously been found.

In this paper we give an algorithm to count closed tree-like walks in a regular graph up to a given length. For small m, this provides expressions for the number of m-matchings in the graph in terms of the numbers of copies of certain small subgraphs that appear in the graph. The simplest of these expressions were already known, having been rediscovered by numerous authors using ad hoc methods. We offer the first general method for producing the expressions. We also find generating functions that isolate the contribution from the simplest kind of subgraph – namely a single cycle of arbitrary length.

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Type
Paper
Information
Combinatorics, Probability and Computing , Volume 19 , Issue 3 , May 2010 , pp. 463 - 480
Copyright
Copyright © Cambridge University Press 2010

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