Constructions of small regular bipartite graphs of girth 6

In this article, some structures in the projective plane of order q are found which allow us to construct small k regular balanced bipartite graphs of girth 6 for all k ≤ q . When k = q , the order of these q-regular graphs is 2(q2−1); andwhen k ≤ q−1, the order of these k -regular graphs is 2(qk − 2). Moreover, the incidence matrix of a k -regular balanced bipartite graph of girth 6 having 2(qk − 2) vertices, where k is an integer and q is a prime power with 3 ≤ k ≤ q − 1, is provided. These graphs improve upon the best known upper bounds for the number of vertices in regular graphs of girth 6. © 2010 Wiley Periodicals, Inc. NETWORKS, Vol. 57(2), 121–127 2011