A Brouwer fixed point theorem for graph endomorphisms

We prove a Lefschetz formula L(T ) = ∑ x∈F iT (x) for graph endomorphisms T : G→ G, where G is a general finite simple graph and F is the set of simplices fixed by T . The degree iT (x) of T at the simplex x is defined as (−1)dim(x)sign(T |x), a graded sign of the permutation of T restricted to the simplex. The Lefschetz number L(T ) is defined similarly as in the continuum as L(T ) = ∑ k(−1)tr(Tk), where Tk is the map induced on the k’th cohomology group Hk(G) of G. A special case is the identity map T , where the formula reduces to the Euler-Poincaré formula equating the Euler characteristic with the cohomological Euler characteristic. The theorem assures that if L(T ) is nonzero, then T has a fixed clique. A special case is the discrete Brouwer fixed point theorem for graphs: if T is a graph endomorphism of a connected graph G, which is star-shaped in the sense that only the zero’th cohomology group is nontrivial, like for connected trees or triangularizations of star shaped Euclidean domains, then there is clique x which is fixed by T . If A is the automorphism group of a graph, we look at the average Lefschetz number L(G). We prove that this is the Euler characteristic of the graph G/A and especially an integer. We also show that as a consequence of the Lefschetz formula, the zeta function ζT (z) = exp( ∑∞ n=1 L(T n) z n n ) is a product of two dynamical zeta functions and therefore has an analytic continuation as a rational function. This explicitly computable product formula involves the dimension and the signature of prime orbits.