We have seen in [17] that averaging the Poincaré-Hopf index theorem formula [25]

\sum _{x\in V}{i}_{f}(x)=\chi (G)

over a probability space of all injective functions f:V\to R leads to Gauss-Bonnet [26]

\sum _{x\in V}K(x)=\chi (G).

It is therefore natural to look at the average {i}_{T}(x) as a curvature when we sum up over all stabilizer elements in {\mathcal{A}}_{x}={Aut}_{x}(G).

**Definition** Define the Lefschetz curvature of a simplex x\in \mathcal{G} as

\kappa (x)=\frac{1}{|\mathcal{A}|}\sum _{T\in {\mathcal{A}}_{x}}{i}_{T}(x)

and the average Lefschetz number

L(G)=\frac{1}{|\mathcal{A}|}\sum _{T\in \mathcal{A}}L(T)

when averaging over all automorphisms.

The number L(G) has an interpretation as the expected index of fixed points of a graph if we chose a random automorphism in the automorphism group. It is a lower bound for the expected number of fixed points of a random automorphism on a graph.

**Examples** (1) For a cycle graph {C}_{n} with n\ge 4, half of the automorphisms have L(T)=0 and half have L(T)=2. The average Lefschetz number is 1.

(2) For a complete graph {K}_{n}, all automorphisms satisfy L(T)=1 so that the average Lefschetz number is 1.

(3) For the Petersen graph *G*, the average Lefschetz number is 1.

If the Lefschetz formula is compared with the Poincaré-Hopf formula, then \kappa (x) is an analogue of Euler curvature and the next result is an analogue of Gauss-Bonnet but where we sum over all simplices in *G*. The Lefschetz curvature is a nonlocal property. It does not depend only on a small neighborhood of the point, but on the symmetries which fix the point = simplex.

**Theorem 5.1** (Average Lefschetz)

\sum _{x\in \mathcal{G}}\kappa (x)=L(G).

*Proof* Use the Lefschetz fixed-point theorem to sum over \mathcal{A}=Aut(G):

\begin{array}{rcl}L(G)& =& \frac{1}{|\mathcal{A}|}\sum _{T\in \mathcal{A}}L(T)\\ =& \frac{1}{|\mathcal{A}|}\sum _{T\in \mathcal{A}}\sum _{x\in \mathcal{F}(T)}{i}_{x}(T)\\ =& \frac{1}{|\mathcal{A}|}\sum _{x\in \mathcal{G}}\sum _{T\in {\mathcal{A}}_{x}}{i}_{x}(T)\\ =& \sum _{x\in \mathcal{G}}\frac{1}{|\mathcal{A}|}\sum _{T\in {\mathcal{A}}_{x}}{i}_{x}(T)=\sum _{x\in \mathcal{G}}\kappa (x).\end{array}

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**Remark** Unlike the Gauss-Bonnet theorem, this Gauss-Bonnet type theorem sums over all possible simplices \mathcal{G}, not only vertices *V* of the graph. The Lefschetz curvature is constant on each orbit of the automorphism group \mathcal{A} and the sum over all curvatures over such an equivalence class is an integer 1 or −1. Theorem 5.2 is the Euler-Poincaré formula in disguise since we will interpret L(G) as an Euler characteristic of an ‘orbifold’ chain.

**Examples** (1) If *G* is the complete graph {K}_{n+1}, then \mathcal{A}={S}_{n+1} is the full permutation group and since L(T)=1 for all *T*, we also have L(G)=1. Now lets compute the Lefschetz curvature. For every fixed *x* we have {i}_{T}(x)={(-1)}^{dim(x)}sign(T|x) and averaging over all *T* gives zero except if *x* is a vertex, where

\kappa (x)=\frac{|{\mathcal{A}}_{x}|}{|\mathcal{A}|}=\frac{1}{n+1}.

The Lefschetz curvature of a vertex is the same than the Euler curvature of a vertex. The curvature is zero on {\mathcal{G}}_{k} for k>0 because the indices of even odd dimensional permutations cancel.

(2) If *G* is star-shaped then L(T)=1 for all *T* and L(G)=1. It reflects the Brouwer analogue that every transformation has a fixed point. If *G* is a star graph {S}_{n}, then the automorphism group is {D}_{n}. For the center point {i}_{T}(x)=1 for all transformations and \kappa (x)=1. All other points have \kappa (x)=0. While the Euler curvature is positive at the spikes and negative in the center, the Lefschetz curvature is entirely concentrated at the center.

(3) If G={C}_{n} for n\ge 4, then \mathcal{A}={D}_{2n} is the dihedral group. For reflections we have L(T)=2, for the rotations, L(T)=0. Therefore, L(G)=1. The stabilizer group {\mathcal{A}}_{x}(G) consists always of two elements whether it is a vertex or edge and {i}_{T}(x)=1 in both cases. We have \kappa (x)=1/(2n) and {\sum}_{x}\kappa (x)=1. The curvature is located both on vertices and edges. Unlike the Euler curvature, the Lefschetz curvature is now nonzero.

(4) If *G* is the wheel graph {W}_{n} with n\ge 4, then again \mathcal{A} is the dihedral group. We still have L(G)=1 but now L(T)=1 for all automorphisms. The center vertex has the full automorphism group as stabilizer group and {i}_{T}(x)=1 for any transformation. Therefore, \kappa (x)=1 at the center and \kappa (x)=0 everywhere else. The center has grabbed all curvature.

(5) If *G* has a trivial automorphism group, then L(G)=\chi (G) is the Euler characteristic. Also each stabilizer group is trivial and {i}_{T}(x)={(-1)}^{|x|} so that \kappa (x)={i}_{T}(x)={(-1)}^{|x|}. In this case, the curvature is spread on all simplices, even-dimensional ones have positive curvature and odd-dimensional ones have negative curvature. It is amusing that the Euler-Poincaré formula can now be seen as a Gauss-Bonnet formula for Lefschetz curvature.

(6) For the octahedron *G*, the orientation preserving automorphisms *T* satisfy L(T)=2. They are realized as rotations if the graph is embedded as a convex regular polygon. The orientation reversing automorphisms have L(T)=0. The average Lefschetz number is L(G)=1 and the Lefschetz curvature is constant 1 at every point.

(7) We can look at the Erdoes-Rényi probability space {\mathrm{\Omega}}_{n} [27] of {2}^{m} graphs *G* on a vertex set with m=n(n-1)/2 edges. The number L(G) is a random variable on {\mathrm{\Omega}}_{n}. We computed the expectation {\mathrm{E}}_{n}[L] for small *n* as follows: {\mathrm{E}}_{2}[L]=1, {\mathrm{E}}_{3}[L]=11/8, {\mathrm{E}}_{4}[L]=43/32, {\mathrm{E}}_{5}[L]=1\text{,}319/1\text{,}024, {\mathrm{E}}_{6}[L]=8\text{,}479/8\text{,}192. Like the expectation of Euler characteristic of random graphs, the expectation of L(G) is expected to oscillate more and more as n\to \mathrm{\infty}. While L(G) takes values 1 or 2 in the case n=1,\dots ,5, there are graphs on 6 vertices, where the maximal Lefschetz number is 3 and the minimal Lefschetz number is 0. The computation for n=6 is already quite involved since we have 32,768 graphs and look at all the automorphisms and for each automorphism find all fixed points.

We will now show that the average Lefschetz number L(G) obtained by averaging over the automorphism group \mathcal{A}=Aut(G) is always an integer, since it is the Euler characteristic of a chain, which is an integer.

**Definition** Let G/\mathcal{A} be the orbigraph (a chain) defined by the automorphism group \mathcal{A} acting on *G*. Two vertices are identified if there is an automorphism mapping one into the other.

**Remarks** (1) G/A is not a graph in general. It sometimes is a multigraph, possibly with loops [28] but in general, it is only a chain.

(2) If geometric graphs *G* in which unit spheres have topological properties from spheres and fixed dimension are considered discrete analogues of manifolds and ℬ is a subgroup of automorphisms of *G*, then G/\mathcal{B} plays the role of an orbifold. Examples are geometric graphs with boundary, where each unit sphere is either sphere-like or a half-sphere of the same fixed dimension.

**Theorem 5.2** (Average Lefschetz is Euler characteristic)

*The Lefschetz number satisfies* L(G)=\chi (G/\mathcal{A}) *and is an integer*.

*Proof* The proof uses elementary group theory and Theorem 3.1.

(1) The Burnside lemma for the finite group \mathcal{A} acting on \mathcal{G} assures

|\mathcal{G}/\mathcal{A}|=\frac{1}{|\mathcal{A}|}\sum _{T\in \mathcal{A}}\left|{\mathcal{F}}^{T}\right|,

where {\mathcal{F}}^{T} is the set of fixed points of *T* and \mathcal{G}/\mathcal{A} is the set of simplices in G/\mathcal{A}.

(2) The number {(-1)}^{dim(x)} of a simplex x\in G/\mathcal{A} is equal to the index {i}_{T}(y) for every simplex *y* which projects onto *x*: we have seen in the Introduction that the dimension dim(y)=|y|-1 of the simplex of T|y satisfies

{i}_{T}(y)={(-1)}^{|y|-1}.

(3) Let {\mathcal{G}}_{+} be the set of simplices *x*, which are mapped under \mathcal{G}\to \mathcal{G}/\mathcal{A} to an even dimensional simplex. These are the simplices *y* for which T|y have index 1 independent of *T*. Similarly, let {\mathcal{G}}_{-} be the set of simplices which are projected to an odd dimensional simplex. All these simplices have negative index for all T\in \mathcal{A}. We therefore know that we have a partition \mathcal{G}={\mathcal{G}}_{+}\cup {\mathcal{G}}_{-} and that for any T\in \mathcal{A} and every y\in \mathcal{G} the index {i}_{y}(T) is equal to {(-1)}^{dim(x)} where x=y/\mathcal{A}.

(4) We can now use the Burnside lemma restricted to \mathcal{A} invariant sets {\mathcal{G}}_{+}, {\mathcal{G}}_{-} and get

\begin{array}{c}{|\mathcal{G}/\mathcal{A}|}_{2k}=\frac{1}{|\mathcal{A}|}\sum _{T\in \mathcal{A}}\left|{\mathcal{F}}_{+}^{T}\right|,\hfill \\ {|\mathcal{G}/\mathcal{A}|}_{2k+1}=\frac{1}{|\mathcal{A}|}\sum _{T\in \mathcal{A}}\left|{\mathcal{F}}_{-}^{T}\right|,\hfill \end{array}

where {\mathcal{F}}_{\pm}^{T} is the set of fixed simplices *y* of *T* for which {i}_{T}(y)=\pm 1.

(5) Let now {|\mathcal{G}/\mathcal{A}|}_{k} the set of simplices in \mathcal{G}/\mathcal{A} which have dimension *k*. We use the Lefschetz fixed point formula to finish the proof:

\begin{array}{rcl}\chi (\mathcal{G}/\mathcal{A})& =& \sum _{k=0}^{\mathrm{\infty}}{(-1)}^{k}{|\mathcal{G}/\mathcal{A}|}_{k}=\frac{1}{|\mathcal{A}|}\sum _{T\in \mathcal{A}}\left|{\mathcal{F}}_{+}^{T}\right|-\left|{\mathcal{F}}_{-}^{T}\right|\\ =& \frac{1}{|\mathcal{A}|}\sum _{T\in \mathcal{A}}\sum _{x\in \mathcal{F}(T)}{i}_{T}(x)=\frac{1}{|\mathcal{A}|}\sum _{T\in \mathcal{A}}L(T)=L(G).\end{array}

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**Remarks** (1) Since L(G)=\chi (G/\mathcal{A}) and *κ* is constant on each orbit, the Lefschetz curvature of a simplex *x* can be rewritten as {(-1)}^{|x/\mathcal{A}|}/|\mathcal{A}x| where x/\mathcal{A} is the simplex after identification with \mathcal{A} and \mathcal{A}x is the orbit of *x* under the automorphism group. Since L(G)=\chi (G/\mathcal{A}), the Gauss-Bonnet type formula (Theorem 5.1) can also be seen to an Euler-Poincaré formula in general. The number \kappa (x) encodes so the orbit length of *x* under the automorphism group \mathcal{A}.

(2) One can also see this as graded summation of an elementary result in linear algebra (see [29], p.21): if a finite group acts linearly on a finite dimensional vector space *V*, then dim(\mathcal{F})=(1/|\mathcal{A}|){\sum}_{T\in \mathcal{A}}tr(T). Let j:\mathcal{F}\to V be the inclusion. Define f(v)=1/|\mathcal{A}|{\sum}_{T\in A}T(v). The image of *f* is in ℱ. If \pi :V\to \mathcal{F} is the projection then f=j\pi. If v\in \mathcal{F}, then Tv=v for all T\in A so that f(v)=v. Therefore \pi j=\mathrm{Id}|\mathcal{F} and dim(F)=tr(\mathrm{Id}|\mathcal{F})=tr(\pi j)=tr(j\pi )=tr(f). If \mathcal{A} is cyclic this simplifies to dim(\mathcal{F})=tr(T).

(3) The proof Theorem 5.1 does not require G/A to be a graph. If *A* is a group acting as automorphism on *G*, then G/A is only a chain, not a graph.

**Examples** (1) Let *G* be the complete graph {K}_{n}. Its automorphism group has *n*! elements. The orbifold graph is a single point. The average Lefschetz number is 1.

(2) Let *G* be cycle graph {C}_{n}. The automorphism group is the dyadic group {D}_{n} with 2*n* elements. The orbigraph is again a single point. The average Lefschetz number is 1.

(3) Let *G* be the discrete graph {P}_{n}. Its automorphism group is the full permutation group again. The orbifold graph is a single point. The average Lefschetz number is 1.

(4) Let *G* be the octahedron. Its automorphism group has 48 elements. The orbigraph is again a single point and the average Lefschetz number is 1.

**Remark** The analogue statement for manifolds needs more algebraic topology like the Leray-Serre spectral sequence [30]: if a manifold *G* has a finite group \mathcal{A} of symmetries, then the average Lefschetz number L(T) of all the symmetry transformations *T* is the Euler characteristic \chi (O) of the orbifold O=M/\mathcal{A}.