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Average Degree in Graph Powers


  • Contract grant sponsors: NSERC Discovery Grant and a Sloan Fellowship, NSERC Postdoctoral Fellowship.


The kth power of a simple graph G, denoted by math formula, is the graph with vertex set math formula where two vertices are adjacent if they are within distance k in G. We are interested in finding lower bounds on the average degree of math formula. Here we prove that if G is connected with minimum degree math formula and math formula, then G4 has average degree at least math formula. We also prove that if G is a connected d-regular graph on n vertices with diameter at least math formula, then the average degree of math formula is at least

display math

Both these results are shown to be essentially best possible; the second is best possible even when math formula is arbitrarily large.