Dr. Patrick Bahls, 16 April 2007
  Asymptotic connectivity and the
geometry of a graph

ABSTRACT: In 2003 Beineke, Oellermann, and Pippert defined the average connectivity ĸ(G)
of a finite graph G in order to provide a useful measure for the overall structural integrity of the graph. Roughly speaking, ĸ measures the expected number of edges that must be removed in order to disconnect the graph.

To generalize this measure to infinite graphs G, we must define asymptotic connectivity ĸ_a(G) by considering the limiting average connectivity of members in a family of increasingly large subgraphs of G. In passing to the limit, the topological measure provided by connectivity becomes a growth-related geometric statistic that records information about the curvature of the graph. In the case of one-ended planar graphs without cut vertices, should ĸ_a(G) exist, it is less than 2 if and only if G is hyperbolic in the sense of Gromov.

The talk will assume only minimal knowledge of graph theory and group theory and will be accessible to undergraduates.


BIOGRAPHICAL SKETCH:  Patrick Bahls received his B.S. in Mathematics from the University of Denver before undertaking doctoral study at Vanderbilt University, where he received his Ph.D. in 2002, working with Prof. Mike Mihalik in the field of geometric group theory.  Following this he moved to Illinois to serve three years as a postdoctoral scholar at the University of Illinois, Urbana-Champaign, working with Ilya Kapovich and Paul Schupp on problems from combinatorial group theory.  For two years now he has been a member of the faculty at the University of North Carolina, Asheville, where he divides his time between research in group theory and graph theory, developing inquiry-based learning courses, assorted vegetarian cookery, and organizing pie-eating contests for the UNCA Math Club.