INCIDENCE MATRICES OF PROJECTIVE PLANES AND OF SOME REGULAR BIPARTITE GRAPHS OF GIRTH 6 WITH FEW VERTICES

摘要

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~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,... ,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 ≠ 0. Using these quasi row-disjoint Latin squares we find (q - l)-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.