This paper describes the construction of all the vertex-transitive graphs on 24 vertices, thus extending the currently available catalogues. This construction differs significantly from previous constructions of the vertex-transitive graphs of order up to 23 in that we are forced to use far more sophisticated group-theoretic techniques. We include an analysis of all the symmetric graphs on 24 v...

We prove that a tournament with n vertices has more than 0.13n(1 + o(1)) edge-disjoint transitive triples. We also prove some results on the existence of large packings of k-vertex transitive tournaments in an n-vertex tournament. Our proofs combine probabilistic arguments and some powerful packing results due to Wilson and to Frankl and Rödl.

The k-dimensional Weisfeiler-Leman algorithm (k-WL) is a very useful combinatorial tool in graph isomorphism testing. We address the applicability of k-WL to recognition properties. Let G be an input with n vertices. show that, if prime, then vertex-transitivity can seen straightforward way from output 2-WL on and vertex-individualized copies G. This perhaps first non-trivial example using for ...

we find recursive formulae for the number of perfect matchings in a graph g by splitting g into subgraphs h and q. we use these formulas to count perfect matching of p hypercube qn. we also apply our formulas to prove that the number of perfect matching in an edge-transitive graph is , where denotes the number of perfect matchings in g, is the graph constructed from by deleting edges with an en...