  A finite abstract simplicial complex G defines the Barycentric refinement
graph phi(G) = (G,{ (a,b), a subset b or b subset a }) and the connection graph
psi(G) = (G,{ (a,b), a intersected with b not empty }). We note here that both
functors phi and psi from complexes to graphs are invertible on the image
(Theorem 1) and that G,phi(G),psi(G) all have the same automorphism group and
that the Cartesian product of G corresponding to the Stanley-Reisner product of
phi(G) and the strong Shannon product of psi(G), have the product automorphism
groups. Second, we see that if G is a Barycentric refinement, then phi(G) and
psi(G) are graph homotopic (Theorem 2). Third, if gamma is the geometric
realization functor, assigning to a complex or to a graph the geometric
realization of its clique complex, then gamma(G) and gamma(phi(G)) and
gamma(psi(G)) are all classically homotopic for a Barycentric refined
simplicial complex G (Theorem 3). The Barycentric assumption is necessary in
Theorem 2 and 3. There is compatibility with Cartesian products of complexes
which manifests in the strong graph product of connection graphs: if two graphs
A,A' are homotopic and B,B' are homotopic, then A . B is homotopic to A' . B'
(Theorem 4) leading to a commutative ring of homotopy classes of graphs.
Finally, we note (Theorem 5) that for all simplicial complexes G as well as
product G=G_1 x G_2 ... x G_k, the Shannon capacity Theta(psi(G)) of psi(G) is
equal to the number m of zero-dimensional sets in G. An explicit Lowasz
umbrella in R^m leads to the Lowasz number theta(G) leq m and so
Theta(psi(G))=theta(psi(G))=m making Theta compatible with disjoint union
addition and strong multiplication.
