Constructions of small regular bipartite graphs of girth 6

Abstract

In this article, some structures in the projective plane of order $q$ **image** are found which allow us to construct small $k$ **image** -regular balanced bipartite graphs of girth 6 for all $k\le q$ **image** . When $k=q$ **image** , the order of these $q$ **image** -regular graphs is $2(q^2-1)$ **image** ; and when $k\le q-1$ **image** , the order of these $k$ **image** -regular graphs is $2(qk-2)$ **image** . Moreover, the incidence matrix of a $k$ **image** -regular balanced bipartite graph of girth 6 having $2(qk-2)$ **image** vertices, where $k$ **image** is an integer and $q$ **image** is a prime power with $3\le k\le q-1$ **image** , 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 © 2011 Wiley Periodicals, Inc.