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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(q^2−1); and when 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.
CitationAraujo, M.; Balbuena, C. Constructions of small regular bipartite graphs of girth 6. "Networks", Març 2011, vol. 57, núm. 2, p. 121-127.
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