9 - p-Ranks

Summary

In this short chapter we discuss the p-ranks of matrices related to (strongly regular) graphs. The p-rank of an integral matrix is the rank over the finite field of order p. Designs or graphs with the same parameters can sometimes be distinguished by considering the 𝑝-rank of associated matrices. For strongly regular graphs the interesting primes p are those dividing r-s (where r and s are the eigenvalues distinct from the valence), otherwise the p-rank is completely determined by the parameters. We list the interesting p-ranks of many graphs and discuss also some families of graphs, such as triangular graphs, Paley and Peisert graphs, symplectic graphs. We also discuss the Smith normal form of the adjacency matrix of some families of graphs, such as the complete graphs, lattice graphs, triangular graphs, Paley and Peisert graphs.