Invariant Projections on Cycle Spaces of Polyhedral Graphs: An Orbit-Count Formula and a Characterisation of the Regular Solids

Abstract

Abstract Let Γ be the skeleton of a convex polyhedron, with cycle space H₁(Γ) and cut space K(Γ), and set c(Γ) = dim Hom_{Aut(Γ)}(H₁(Γ), K(Γ)). By a theorem of Dilworth, Kutzarova and Ostrovskii, c(Γ) = 0 is equivalent to uniqueness of the invariant projection of the signed edge space onto H₁(Γ). We determine c for every convex polyhedron. First, we prove the orbit formula c(Γ) = N(F × V) − N(F) − N(V) + δ, where N counts automorphism-group orbits whose stabiliser contains no orientation-reversing element and δ is the chirality correction. We then prove duality invariance and show that c(Γ) = 0 if and only if Γ is the graph of a Platonic solid. The converse reduces vanishing to a covering condition by mirror planes, forcing vertex directions and face axes onto finitely many rotation axes; a finite forced-ratio enumeration closes the polyhedral reflection groups and uniform arguments settle the axial families. We also give closed formulas for chiral toroidal maps and counterexamples showing that the spherical characterisation does not extend unchanged to the torus or projective plane. Keywords: Cycle space; cut space; invariant projection; equivariant map; polyhedral graph; regular polyhedron; orbit counting; chirality; regular map; flag-transitivity; transportation cost space. Mathematics Subject Classification (2020): 05E18 (primary); 05C10, 05C25, 20C15, 52B15, 46B85 (secondary).