Cycles, Paths, Connectivity and Diameter in Distance Graphs

Abstract

For n ? ? and D ? ?, the distance graph $P_n^D$ has vertex set { 0,1,...,n ? 1} and edge set { ij|0 ≤ i,j ≤ n ? 1, |j ? i| ? D}. The class of distance graphs generalizes the important and very well-studied class of circulant graphs which have been proposed for numerous applications concerning networks, distributed systems and chip design. We prove that the class of circulant graphs coincides with the class of regular distance graphs. Extending some of the fundamental results concerning circulant graphs, we study the existence of long cycles and paths in distance graphs and analyse the computational complexity of problems related to their connectivity and diameter.