Contract grant sponsors: NSERC Discovery Grant and a Sloan Fellowship, NSERC Postdoctoral Fellowship.
Average Degree in Graph Powers
Article first published online: 8 MAY 2012
© 2012 Wiley Periodicals, Inc.
Journal of Graph Theory
Volume 72, Issue 1, pages 7–18, January 2013
How to Cite
DeVos, M., McDonald, J. and Scheide, D. (2013), Average Degree in Graph Powers. J. Graph Theory, 72: 7–18. doi: 10.1002/jgt.21628
- Issue published online: 23 OCT 2012
- Article first published online: 8 MAY 2012
- Manuscript Revised: 17 NOV 2011
- Manuscript Received: 14 DEC 2010
- NSERC Discovery Grant
- NSERC Postdoctoral Fellowship
- graph power;
- average degree
The kth power of a simple graph G, denoted by , is the graph with vertex set 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 . Here we prove that if G is connected with minimum degree and , then G4 has average degree at least . We also prove that if G is a connected d-regular graph on n vertices with diameter at least , then the average degree of is at least
Both these results are shown to be essentially best possible; the second is best possible even when is arbitrarily large.