Connectedness, Classes and Cycle Index

  • Authors:

  • E. A. Bender;P. J. Cameron;A. M. Odlyzko;L. B. Richmond

  • Affiliations:

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

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 1999

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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.