Determinants of Laplacians on graphs

Abstract

IN RECENT years, it has been observed that the determinant of a Laplacian on a manifold can often be expressed in terms of the closed orbits of a flow on that (or a related) manifold. Examples include the work of many authors (see, for example [2], [3], [6], [14], [16], [19], [20]) which uses the Selberg trace formula to express the determinant of a Laplacian acting on sections of a vector bundle over a Riemann surface in terms of the closed geodesics on the surface. Other examples can be found in the work of D. Fried relating Reidemeister Torsion (a combinatorial invariant of finite C-W complexes which can be expressed in terms of determinants of combinatorial Laplacians) to the closed orbits of flows of various types ([S], [6], [7]). In this paper we show that this relationship is, in fact, fundamental, and can be seen on the level of graphs. Let G be a finite graph (all terms will be defined precisely in Section 1) with a weight attached to each vertex and edge. These weights induce an inner product on the spaces V* (the complex functions on the set of vertices of G) and E* (the complex functions on the set of edges). If 6 is the usual coboundary operator from V* to E* we define a (combinatorial) Laplacian A by A = 6*6