Vertex-transitive graphs

Pentavalent symmetric graphs of order 2p 3

“…To do this, we need to introduce the so called coset graph (see [26,33]) constructed from a finite group G relative to a subgroup H of G and a union D of some double cosets of …”

“…By [33], the graph X is isomorphic to a coset graph Cos(A, H, D), where H = A u is the vertex stabilizer of u ∈ V (X) in A and D consists of all elements of A which map u to one of its neighbors. It is easy to show that H A = 1 and that D is a union of some double cosets of H in A satisfying D = D −1 .…”

“…Furthermore, g can be chosen as a 2-element in A, and the valency of X is |D|/|H| = |H : H ∩ H g |. For more details regarding coset graphs, see, for example, [9,24,33]. Now we introduce three pentavalent symmetric graphs of order 6p for some prime p which were constructed in [17, Section 3].…”

Tetravalent one-regular graphs of order 2pq

“…Assume that H = A u is the stabilizer of u ∈ V (X) and that g ∈ A interchanges u and v. It is easy to see that the core H A of H in A (the largest normal subgroup of A contained in H ) is trivial, and that HgH consists of all elements of A which maps u to one of its neighbors in X. By [18,30], the graph X is isomorphic to the coset graph X * = Cos(A, H, HgH ), which is defined as the graph with vertex set V (X * ) = {H a : a ∈ A}, the set of right cosets of H in A, and edge set E(X * ) = {{H a, H da} a ∈ A, d ∈ HgH }. The valency of X * is |HgH |/|H | = |H : H ∩ H g |, and X * is connected if and only if HgH generates A.…”

Large Cayley Graphs and Digraphs with Small Degree and Diameter

“…Sabidussi [37] showed that all vertex transitive graphs can be obtained as Cayley coset graphs of groups relative to some set of generators. Similar results hold for digraphs [27].…”