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Connectedness, Classes and Cycle Index

Published online by Cambridge University Press:  01 January 1999

E. A. BENDER ,
P. J. CAMERON ,
A. M. ODLYZKO  and
L. B. RICHMOND
E. A. BENDER
Affiliation:
Center for Communications Research, 4320 Westerra Court, San Diego, CA 92121, USA (e-mail: ed@ccrwest.org)
P. J. CAMERON
Affiliation:
School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London E1 4NS, England (e-mail: p.j.cameron@qmw.ac.uk)
A. M. ODLYZKO
Affiliation:
AT&T Bell Laboratories, 600 Mountain Avenue, Murray Hill, NJ 07974–0636, USA (e-mail: amo@research.att.com)
L. B. RICHMOND
Affiliation:
Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada (e-mail: lbrichmo@watdragon.uwaterloo.ca)
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Abstract

This paper begins with the observation that half of all graphs containing no induced path of length 3 are disconnected. We generalize this in several directions. First, we give necessary and sufficient conditions (in terms of generating functions) for the probability of connectedness in a suitable class of graphs to tend to a limit strictly between zero and one. Next we give a general framework in which this and related questions can be posed, involving operations on classes of finite structures. Finally, we discuss briefly an algebra associated with such a class of structures, and give a conjecture about its structure.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 8 , Issue 1-2 , January 1999 , pp. 31 - 43
Copyright
© 1999 Cambridge University Press

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