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ASYMPTOTIC ENUMERATION OF SYMMETRIC INTEGER MATRICES WITH UNIFORM ROW SUMS

Part of: Special matrices Combinatorics

Published online by Cambridge University Press:  04 March 2012

BRENDAN D. MCKAY  and
JEANETTE C. MCLEOD
BRENDAN D. MCKAY
Affiliation:
Research School of Computer Science, Australian National University, Canberra ACT 0200, Australia (email: bdm@cs.anu.edu.au)
JEANETTE C. MCLEOD*
Affiliation:
Department of Mathematics and Statistics, University of Canterbury, Christchurch 8140, New Zealand (email: jeanette.mcleod@canterbury.ac.nz)
*
✉For correspondence; e-mail: jeanette.mcleod@canterbury.ac.nzd
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Abstract

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We investigate the number of symmetric matrices of nonnegative integers with zero diagonal such that each row sum is the same. Equivalently, these are zero-diagonal symmetric contingency tables with uniform margins, or loop-free regular multigraphs. We determine the asymptotic value of this number as the size of the matrix tends to infinity, provided the row sum is large enough. We conjecture that one form of our answer is valid for all row sums. An example appears in Figure 1.

Keywords

symmetric matrixasymptotic enumerationcontingency tablemultigraphdegree sequence

MSC classification

Secondary: 05A16: Asymptotic enumeration 15B36: Matrices of integers
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 92 , Issue 3 , June 2012 , pp. 367 - 384
Copyright
Copyright © 2013 Australian Mathematical Publishing Association Inc.

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