Growth of graph powers

For a graph G, its rth power is constructed by placing an edge between two vertices if they are within distance r of each other. In this note we study the amount of edges added to a graph by taking its rth power. In particular we obtain that either the rth power is complete or "many" new edges are added. This is an extension of a result obtained by P. Hegarty for cubes of graphs.Comment: 6 pages, 1 figur

Edge growth in graph powers

For a graph G, its rth power G^r has the same vertex set as G, and has an edge between any two vertices within distance r of each other in G. We give a lower bound for the number of edges in the rth power of G in terms of the order of G and the minimal degree of G. As a corollary we determine how small the ratio e(G^r)/e(G) can be for regular graphs of diameter at least r