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

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, for r ≥ 3, either the rth power is complete or "many" new edges are added. In this direction, Hegarty showed that there is a constant ε > 0 such e(G3) ≥ (1 + ε)e(G). We extend this result in two directions. We give an alternative proof of Hegarty's result with an improved constant of ε = 1/6. We also show that for general

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

Edge growth in graph squares

We resolve a conjecture of Hegarty regarding the number of edges in the square of a regular graph. If $G$ is a connected $d$-regular graph with $n$ vertices, the graph square of $G$ is not complete, and $G$ is not a member of two narrow families of graphs, then the square of $G$ has at least $(2-o_d(1))n$ more edges than $G$

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