Spectral properties of Cayley graphs over Mmxn(Fq)

摘要

Let F-q be the finite field of order q and let M-mxn(F-q) be the additive (abelian) group consisting of all m x n matrices over F-q. 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 M-mxn(F-q), and two vertices A, B & ISIN; M-mxn(F-q) 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].

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