Abstract: For a finite group $$G$$ with subgroups $$H$$ and $$K$$, the coset incidence graph $$\Gamma(G; H, K)$$ joins $$aH$$ to $$bK$$ 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 $$HKH\cdots$$ 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 $$HK = KH$$, 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 $$HK$$; its spectrum is described by principal angles between fixed subspaces and, for Gelfand pairs, by character averages over $$HK$$. The Heawood and Tutte–Coxeter graphs illustrate the results.

Keywords: Bipartite graph, coset geometry, girth, edge ideal, Castelnuovo–Mumford regularity, Cohen–Macaulay graph, Hecke algebra, Gelfand pair, principal angles, generalized polygon

Pages: 571-588