The McKean-Singer Formula in Graph Theory

نویسنده

  • Oliver Knill
چکیده

For any finite simple graphG = (V,E), the discrete Dirac operator D = d+d∗ and the Laplace-Beltrami operator L = dd∗+d∗d = (d+d∗)2 = D2 on the exterior algebra bundle Ω = ⊕Ωk are finite v × v matrices, where dim(Ω) = v = ∑ k vk, with vk = dim(Ωk) denoting the cardinality of the set Gk of complete subgraphs Kk of G. We prove the McKean-Singer formula χ(G) = str(e−tL) which holds for any complex time t, where χ(G) = str(1) = ∑ k(−1)vk is the Euler characteristic of G. The super trace of the heat kernel interpolates so the Euler-Poincaré formula for t = 0 with the Hodge theorem in the real limit t→∞. More generally, for any continuous complex valued function f satisfying f(0) = 0, one has the formula χ(G) = str(ef(D)). This includes for example the Schrödinger evolutions χ(G) = str(cos(tD)) on the graph. After stating some immediate general facts about the spectrum which includes a perturbation result estimating the Dirac spectral difference of graphs, we mention as a combinatorial consequence that the spectrum encodes the number of closed paths in the simplex space of a graph. We give a couple of worked out examples and see that McKean-Singer allows to find explicit pairs of nonisometric graphs which have isospectral Dirac operators.

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عنوان ژورنال:
  • CoRR

دوره abs/1301.1408  شماره 

صفحات  -

تاریخ انتشار 2012