Geodesic
On the $p$-ranks of the adjacency matrices of distance-regular graphs
Journal of Algebraic Combinatorics, Tome 15 (2002) no. 2, pp. 127-149.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: Let $Gamma$ 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 $Gamma$. 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.
Keywords: $p$-rank, distance-regular graph, adjacency matrix, minimal polynomial 
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     author = {Peeters, Ren\'e},
     title = {On the $p$-ranks of the adjacency matrices of distance-regular graphs},
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Peeters, René. On the $p$-ranks of the adjacency matrices of distance-regular graphs. Journal of Algebraic Combinatorics, Tome 15 (2002) no. 2, pp. 127-149. http://geodesic.mathdoc.fr/item/JAC_2002__15_2_a2/