On the p-Ranks of the Adjacency Matrices of Distance-Regular Graphs
نویسنده
چکیده
Let be a distance-regular graph with adjacency matrix A. Let I be the identity matrix and J the all-1 matrix. Let p be a prime. We study the p-rank of the matrices A + bJ − cI for integral b, c and the p-rank of corresponding matrices of graphs cospectral with . Using the minimal polynomial of A and the theory of Smith normal forms we first determine which p-ranks of A follow directly from the spectrum and which, in general, do not. For the p-ranks that are not determined by the spectrum (the so-called relevant p-ranks) of A the actual structure of the graph can play a rôle, which means that these p-ranks can be used to distinguish between cospectral graphs. We study the relevant p-ranks for some classes of distance-regular graphs and try to characterize distance-regular graphs by their spectrum and some relevant p-rank.
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