Spectral properties of Cayley graphs over Mm×n(Fq)

Abstract

Let Fq be the finite field of order q and let Mm×n(Fq) be the additive (abelian) group consisting of all m×n matrices over Fq. Given an integer r with 0≤r≤min⁡{m,n}, the Cayley graph G(m,n,r) is defined as the graph whose vertices are consisting of all the elements of Mm×n(Fq), and two vertices A,B∈Mm×n(Fq) are adjacent if the rank of A−B (denoted by rank(A−B)) is equal to r. In this paper, a recursion relation for the eigenvalues of G(m,n,r) is established; consequently, explicit formulas for all the eigenvalues of G(m,n,1) are exhibited immediately, which is a main result obtained previously in Delsarte (1975) [4].