Table of Contents
International Journal of Combinatorics
Volume 2013, Article ID 347613, 14 pages
http://dx.doi.org/10.1155/2013/347613
Research Article

An Algebraic Representation of Graphs and Applications to Graph Enumeration

Centro de Estruturas Lineares e Combinatórias, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal

Received 23 July 2012; Accepted 25 September 2012

Academic Editor: Xueliang Li

Copyright © 2013 Ângela Mestre. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We give a recursion formula to generate all the equivalence classes of connected graphs with coefficients given by the inverses of the orders of their groups of automorphisms. We use an algebraic graph representation to apply the result to the enumeration of connected graphs, all of whose biconnected components have the same number of vertices and edges. The proof uses Abel’s binomial theorem and generalizes Dziobek’s induction proof of Cayley’s formula.