  A CIS graph is a graph in which every maximal stable set and every maximal
clique intersect. A graph is well-covered if all its maximal stable sets are of
the same size, co-well-covered if its complement is well-covered, and
vertex-transitive if, for every pair of vertices, there exists an automorphism
of the graph mapping one to the other. We show that a vertex-transitive graph
is CIS if and only if it is well-covered, co-well-covered, and the product of
its clique and stability numbers equals its order. A graph is irreducible if no
two distinct vertices have the same neighborhood. We classify irreducible
well-covered CIS graphs with clique number at most 3 and vertex-transitive CIS
graphs of valency at most 7, which include an infinite family. We also exhibit
an infinite family of vertex-transitive CIS graphs which are not Cayley.
