On The Critical Group of Finite Projective Planes

On The Critical Group of Finite Projective Planes
Stuart Shirell, University of Arkansas
September 17, 2009

The critical group of a graph on n vertices is defined to be $Z^n/(Image(M))$, where M is the Laplacian matrix associated to the graph. This is an isomorphism invariant and is largely viewed as relatively robust.

Relatively little is known about the critical group in general, however.  For example the critical groups of complete graphs, bipartite complete graphs, and the n-cube are (mostly) known, but little else is known about these groups or how they reflect the symmetry properties of the underlying graph. We determine completely the critical groups of non-degenerate, finite projective planes and show that all projective planes of a given order have isomorphic critical groups.