  In this paper we introduce a Cayley-type graph for group-subgroup pairs and
present some elementary properties of such graphs, including connectedness,
their degree and partition structure, and vertex-transitivity. We relate these
properties to those of the underlying group-subgroup pair. From the properties
of the group, subgroup and generating set some of the eigenvalues can be
determined, including the largest eigenvalue of the graph. In particular, when
this construction results in a bipartite regular graph we show a sufficient
condition on the size of the generating sets that results on Ramanujan graphs
for a fixed group-subgroup pair. Examples of Ramanujan pair-graphs that do not
satisfy this condition are also provided, to show that the condition is not
necessary.
