Eigenvalues of the adjacency matrix of cubic lattice graphs

Abstract

A cubic lattice graph is defined to be a graph G, whose vertices are the ordered triplets on n symbols, such that two vertices are adjacent if and only if they have two coordinates in common. If n2(x) denotes the number of vertices y, which are at distance 2 from x and A(G) denotes the adjacency matrix of G, then G has the following properties: (P1) the number of vertices is n3. (P2) G is connected and regular. (P3) n2(x) = S(n − I)2. (P4) the distinct eigenvalues of A(G) are − 3, n−2, 2n−3, (n − 1). It is shown here that if n > 7, any graph G (with no loops and multiple edges) having the properties (P3) − (P4) must be a cubic lattice graph. An alternative characterization of cubic lattice graphs has been given by the author (J. Comb. Theory, Vol. 3, No. 4, December 1967, 386-401). © 1969 by Pacific Journal of Mathematics.