## Graph Powers

August 7, 2008

Earlier I had talked about multiplying the adjacency matrix of a graph by itself and how it leads to some interesting results. Multiplying the adjacency matrix by itself essentially corresponds to raising the graph by some integer exponent while maintaining the same number of nodes.

The powers of graphs are interesting structures. The p^th power, G^p of a graph G is obtained by adding edges between nodes in G that are separated by a pathlength (greater than 1 and) less than or equal to p. Specifically, the square of a graph is the graph resulting from joining all nodes that are separated by a pathlength 2. A cube of a graph is obtained by short-circuiting all 2 length and 3 length paths in the graph.

An important consequence of raising the graph to a power is diameter reduction. If G has diameter d, then graph G^p has diameter \ceil{d/p}. There are several other interesting properties of graph powers that we can observe, which are important in the design of optimal topologies. I’d be interested in knowing your ideas.