An index formula for simple graphs

We prove that any odd dimensional geometric graph G = (V,E) has zero curvature everywhere. To do so, we prove that for every injective function f on the vertex set V of a simple graph the index formula 1 2 [1 − χ(S(x))/2 − χ(Bf (x))] = (if (x) + i−f (x))/2 = jf (x) holds, where if (x) is a discrete analogue of the index of the gradient vector field ∇f and where Bf (x) is a graph defined by G and f . The Poincaré-Hopf formula ∑ x jf (x) = χ(G) allows so to express the Euler characteristic χ(G) of G in terms of smaller dimensional graphs defined by the unit sphere S(x) and the ”hypersurface graphs” Bf (x). For odd dimensional geometric graphs, Bf (x) is a geometric graph of dimension dim(G)−2 and jf (x) = −χ(Bf (x))/2 = 0 implying χ(G) = 0 and zero curvature K(x) = 0 for all x. For even dimensional geometric graphs, the formula becomes jf (x) = 1−χ(Bf (x))/2 and allows with PoincaréHopf to write the Euler characteristic of G as a sum of the Euler characteristic of smaller dimensional graphs. The same integral geometric index formula also is valid for compact Riemannian manifolds M if f is a Morse function, S(x) is a sufficiently small geodesic sphere around x and Bf (x) = S(x) ∩ {y | f(y) = f(x) }.