On the \(p\)-ranks of the adjacency matrices of distance-regular graphs
Let \(\Gamma\) be a distance-regular graph with adjacency matrix \(A\). Let \(I\) be the identity matrix, \(J\) the all-1 matrix, and let \(p\) be a prime. The author considers the \(p\)-rank of the matrices \(A+bJ-cI\), where \(b\) and \(c\) are integers, generalizing results by \textit{A. E. Brouwer} and \textit{C. A. van Eijl} [J. Algebr. Comb. 1, No. 4, 329--346 (1992; Zbl 0780.05039)]. Using the minimal polynomial of \(A\), as well as the theory of Smith normal forms it is shown that most of these ranks follow directly from the spectrum. Those ranks that, in general, do not follow from the spectrum only, depend also on the actual structure of the graph and can be used to distinguish between cospectral graphs. The author presents complete results for Hamming and Doob graphs, as well as further results for few other classes of distance-regular graphs, such as Johnson graphs, generalized quadrangles minus a spread, square \(2\)-designs and Taylor graphs.
scientific article; zbMATH DE number 1747858