The McKean-Singer Formula in Graph Theory

For any finite simple graph G=(V,E), the discrete Dirac operator D=d+d* and the Laplace-Beltrami operator L=d d* + d* d on the exterior algebra bundle Omega are finite v times v matrices, where dim(Omega) = v is the sum of the cardinalities v(k) of the set G(k) of complete subgraphs K(k) of G. We prove the McKean-Singer formula chi(G) = str(exp(-t L)) which holds for any complex time t, where chi(G) = str(1)= sum (-1)k v(k) is the Euler characteristic of G. The super trace of the heat kernel interpolates so the Euler-Poincare formula for t=0 with the Hodge theorem in the real limit t going to infinity. More generally, for any continuous complex valued function f satisfying f(0)=0, one has the formula chi(G) = str(exp(f(D))). This includes for example the Schroedinger evolutions chi(G) = str(cos(t D)) on the graph. After stating some general facts about the spectrum of D which includes statements about the complexity, the product of the non-zero eigenvalues as well as a perturbation result estimating the spectral difference of two graphs, we mention as a combinatorial consequence that the spectrum of D encodes the number of closed paths in the simplex space of a graph. McKean-Singer implies that the number of closed paths of length n starting at an even dimensional simplex is the same than the number of closed paths of length n starting at an odd dimensional simplex. We give a couple of worked out examples and see that McKean-Singer allows to find explicit pairs of non-isometric graphs which have isospectral Dirac operators.

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