Incidence Matrices of Projective Planes and of Some Regular Bipartite Graphs of Girth 6 with Few Vertices

Let q be a prime power and r = 0, 1 . . . , q − 3. Using the Latin squares obtained by multiplying each entry of the addition table of the Galois field of order q by an element distinct from zero, we obtain the incidence matrices of projective planes and the incidence matrices of (q− r)-regular bipartite graphs of girth 6 and q − rq− 1 vertices in each partite set. Moreover, in this work two Latin squares of order q − 1 with entries belonging to {0, 1, . . . , q}, not necessarily the same, are defined to be quasi row-disjoint if and only if the cartesian product of any two rows contains at most one pair (x, x) with x 6= 0. Using these quasi row-disjoint Latin squares we find (q−1)-regular bipartite graphs of girth 6 with q − q − 2 vertices in each partite set. Some of these graphs have the smallest number of vertices known so far among the regular graphs with girth 6.