Distance powers of unitary Cayley graphs

摘要

Let G be a graph and let diam(G) denote the diameter of G. The distance power GN of G is the undirected graph with vertex set V(G), in which x and y are adjacent if their distance d(x, y) in G belongs to N, where N is a non-empty subset of {1,2,…,diam(G)}. The unitary Cayley graph is the graph having the vertex set Zn and the edge set {(a,b):a,b∈Zn,gcd(a−b,n)=1}. In this paper, we determine the energies of distance powers of unitary Cayley graphs, and classify all Ramanujan distance powers of unitary Cayley graphs. By the energies of distance powers of unitary Cayley graphs, we construct infinitely many pairs of non-cospectral equienergetic graphs. Moreover, we characterize all hyperenergetic distance powers of unitary Cayley graphs.