Cayley-type graphs for group–subgroup pairs

In this paper we introduce a Cayley-type graph for group–subgroup pairs (G,H)(G,H) and certain subsets S of G  . We present some elementary properties of such graphs, including connectedness, degree and partition structure, and vertex-transitivity, relating these properties with those of the underlying group–subgroup pair. From the properties of the underlying structures, some of the eigenvalues can be determined, including the largest eigenvalue of the graph. We present a sufficient condition on the group–subgroup pair (G,H)(G,H) and the size of S that results on bipartite Ramanujan graphs. Among those Ramanujan graphs there are graphs that cannot be obtained as Cayley graphs. As another application, we propose the use of group–subgroup pair graphs to model linear error-correcting codes.