The Coset Incidence Graph of a Finite Group: Structure, Edge Ideals, and Spectrum

For a finite group G with subgroups H and K , the coset incidence graph Γ ( G ; H , K ) joins a H to b K when the cosets intersect; it is the Levi graph of a rank-two coset geometry. We show that its combinatorial, homological and spectral invariants are governed by products of H and K . Alternating products H K H ⋯ determine distances, diameter and girth. We classify when the graph is Cohen–Macaulay, unmixed or sequentially Cohen–Macaulay, compute the multiplicity of the edge ideal and, when H K = K H , all its Betti numbers, and determine the regularity under a packing condition in G . The biadjacency matrix is the Hecke operator of the double coset H K ; its spectrum is described by principal angles between fixed subspaces and, for Gelfand pairs, by character averages over H K . The Heawood and Tutte–Coxeter graphs illustrate the results.