Analytic torsion for graphs

For any finite simple graph (V,E), the squared analytic torsion is the positive rational number A(G) = ∏ k Det(Lk) k(−1) , where Lk are the blocks of the Hodge Laplacian L = D2 = (d+d∗)2 of the Whitney complex and Det is the pseudo determinant. Torsion A(G) agrees with the super pseudo determinant SDet(D) = ∏ k Det(Dk) (−1) of the Dirac blocks Dk = d ∗ kdk of the Dirac operator D = d + d∗ and is related to the pseudo determinant Det(D) = ± ∏ k Det(Dk) of D. This gives a generalized matrix tree theorem: A(G) is the ratio of rooted spanning trees on even-dimensional simplices divided by the number of rooted spanning trees in odd simplices. In particular, the classical matrix tree theorem rephrases that for graphs without triangles, A(G) is the number of rooted spanning trees in G. For 2-spheres with |F | triangles, torsion is A(G) = |V |/|F |, rephrasing Von Staudt’s theorem that the number of spanning trees in a 2-sphere G and its dual graph G′ agree. We prove in general A(G) = |V | for graphs homotopic to 1 and A(G) = |V |/|V ′| for (2r)-spheres and A(G) = |V ||V ′| for (2r + 1)-spheres, where V ′ is the set of maximal simplices in G and |V |, |V ′| are the cardinalities G or G′. Torsion, as the super pseudo determinant of the Dirac operator D = d + d∗ can be defined for any bounded differential complex. Similar formulas hold so for Wu torsion of spheres in the Wu complex. We also start to look at the expectation of A on Erdoes-Renyi probability spaces or look into the problem which graphs on n vertices maximize or minimize A(G). The limit limn→∞ A(Gn) for Barycentric refinements of even dimensional spheres can be computed.