A combinatorial proof of the Rayleigh formula for graphs

Abstract

Rayleigh monotonicity in Physics has a combinatorial interpretation. In this paper we give a combinatorial proof of the Rayleigh formula using the Jacobi Identity and the all-minors matrix tree Theorem. Motivated by the fact that the edge set of each spanning tree of G is a basis of the graphic matroid induced by G, we define the Rayleigh monotonicity of the generating polynomial for the set of bases of a matroid and suggest a few related problems. (C) 2007 Elsevier B.V. All rights reserved.