On the p-Rank of the Adjacency Matrices of Strongly Regular Graphs

Let F be a strongly regular graph with adjacency matrix A. Let I be the identity matrix, and J the all-1 matrix. Let p be a prime. Our aim is to study the p-rank (that is, the rank over Fp, the finite field with p elements) of the matrices M = aA + bJ + cI for integral a, b, c. This note is based on van Eijl [8]. 1. The Smith normal form Let us write M ~ N for integral matrices M and N of order n, if there are integral matrices P and Q of determinant ±1 such that N = PMQ. Clearly, ~ is an equivalence relation. Given the matrix M, we can find a diagonal matrix S(M) with S(M) ~ M and S(M) = diag (s1,···,sn) with s1|s2| ···|sn. The matrix S(M) is uniquely determined up to the signs of the Si (we might require Si > 0, but that is not always convenient), and is called the (more precisely, a) Smith normal form of M. (Thus, "S(M) = A" is to be interpreted as stating that A is a Smith normal form of M.) Clearly, Pisi = det S(M) = ±det M, and more generally Pi=1si is the g.c.d. of all minors of M of order k. The p-rank rkp(M) of M equals the number of Si not divisible by p, and the Q-rank rk(M) of M equals the number of nonzero Si. In particular, rkpM < rkM. If p||det M, then rkpM > n e. SNFO. Let M be an integral matrix of order 2, and g the g.c.d. of its four elements. Then S(M) = diag (g,(detM)/g). SNF1. S(Jn + cIn) = diag (1,c ,c(c + n)) for c E Z. (We use exponents to denote multiplicities.) More generally S(bJn + cIn) = diag (g,c ,(bn + c)c/g), where g = gcd(b, c). SNF2. Let A be the adjacency matrix of the n x n grid graph. Then