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

Let $q$ be a prime power and $r=0,1\ldots, q-3$. 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 $q^2-rq-1$ vertices in each partite set. Moreover, in this work two Latin squares of order $q-1$ with entries belonging to $\{0,1,\ldots, q\}$, 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 $x\ne 0$. Using these quasi row-disjoint Latin squares we find $(q-1)$-regular bipartite graphs of girth 6 with $q^2-q-2$ 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.