Eigenvalues and separation in graphs

Abstract

Consider a one-to-one map f:V(G)→V(H) of Graphs G and H, where |G|⩽|H|. We let |f|=min{distH(f(x), f(y)):xyεE(G)}, where distH denotes distance in H. Now define sep(G,H), the separation of G into H, to be the maximum of |f| over all such maps f. Using the Kronecker product of matrices, we develop a method for computing, in certain favorable cases, the set of eigenvalues of graphs of the form (G×H)(k). Here G×H refers to the usual graph product, and S(k) (for a graph S) is the graph obtained from S by joining two points of V(S) by an edge if and only if they are at distance at most k in S. Let Q(n) and Cn× Cn denote the n-dimensional cube and the n × n “discrete torus” respectively, and λmin(S) the smallest eigenvalue of a graph S. We apply our method to analyze λmin(Q(n)(k)) and λmin((Cn× Cn)(k)), obtaining exact values for certain k and asymptotically optimal lower bounds for others. Combining these results with one of the results of Alon and Milman, we obtain bounds for the edge isoperimetric problem in the graphs Q(n)(k)c and (Cn× Cn)(k)c, where Sc denotes the graph obtained from a graph S by joining two vertices if and only if they are not joined in S. As a corollary we obtain functions b(k, p)[c(k, p)] such that if a graph G on p points and q edges satisfies q > b(k, p)[q > c(k, p)], then sep(G, Q(n))⩽ k [sep(G, Cn × Cn⩽ k].