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Keywords:

  • graph power;
  • average degree

Abstract

The kth power of a simple graph G, denoted by inline image, is the graph with vertex set inline image 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 inline image. Here we prove that if G is connected with minimum degree inline image and inline image, then G4 has average degree at least inline image. We also prove that if G is a connected d-regular graph on n vertices with diameter at least inline image, then the average degree of inline image is at least

  • display math

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