In this paper we introduce a Cayley-type graph for group-subgroup pairs (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) 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.
Reyes-Bustos, C. (2016). Cayley-type graphs for group-subgroup pairs. Linear Algebra and Its Applications, 488, 320–349. https://doi.org/10.1016/j.laa.2015.09.049