A classification of pentavalent symmetric graphs of order twice a prime square is given. It is proved that such a graph is a coset graph of Z3.A6 （non-split extension）, or a bi-coset graph of an extra-special group of order 125, or the standard double cover of a specific abelian Cayley digraph of order a prime square.
In this paper, we consider a general composite convex optimization problem with a cone-convex system in locally convex Hausdorff topological vector spaces. Some Fenchel conjugate transforms for the composite convex functions are derived to obtain the equivalent condition of the Stable Farkas Lemma, which is formulated by using the epigraph of the conjugates for the convex functions involved and turns out to be weaker than the classic Slater condition. Moreover, we get some necessary and sufficient conditions for stable duality results of the composite convex functions and present an example to illustrate that the monotonic increasing property of the outer convex function in the objective function is essential. Our main results in this paper develop some recently results.
