The complementary prism of a graph Γ is the ΓΓ̄, which formed from union and its complement Γ̄ by adding an edge between each pair identical vertices in Γ̄. Vertex-transitive self-complementary graphs provide vertex-transitive prisms. It was recently proved author that ΓΓ̄ core, i.e. all endomorphisms are automorphisms, whenever vertex-transitive, self-complementary, either core or complete graph. ...

For two vertices u and v of a graph G, the closed interval I[u, v] consists of u, v, and all vertices lying in some u − v geodesic in G. If S is a set of vertices of G, then I[S] is the union of all sets I[u, v] for u, v ∈ S. If I[S] = V (G), then S is a geodetic set for G. The geodetic number g(G) is the minimum cardinality of a geodetic set. A set S of vertices in a graph G is uniform if the ...

A decomposition of a graph is a partition of the edge set, giving a set of subgraphs. A transitive decomposition is a decomposition which is highly symmetrical, in the sense that the subgraphs are preserved and transitively permuted by a group of automorphisms of the graph. This paper describes some ‘product’ constructions for transitive decompositions of graphs, and shows how these may be used...

A transitive decomposition of a graph is a partition of the edge or arc set giving a set of subgraphs which are preserved and permuted transitively by a group of automorphisms of the graph. In this paper we give some background to the study of transitive decompositions and highlight a connection with partial linear spaces. We then describe a simple method for constructing transitive decompositi...

A finite graph X is half-arc-transitive if its automorphism group is transitive on vertices and edges, but not on arcs. When X is tetravalent, the automorphism group induces an orientation on the edges and a cycle of X is called an alternating cycle if its consecutive edges in the cycle have opposite orientations. All alternating cycles of X have the same length and half of this length is calle...

Let be a finite G-symmetric graph whose vertex set admits a nontrivial G-invariant partition B. It was observed that the quotient graph B of relative to B can be (G,2)-arc transitive even if itself is not necessarily (G,2)-arc transitive. In a previous article of Iranmanesh et al., this observation motivated a study of G-symmetric graphs ( ,B) such that B is (G,2)-arc transitive and, for blocks...

A transitive signature scheme allows a signer to publish a graph in an authenticated and cost-saving manner. The resulting authenticated graph is indeed the transitive closure of the graph constructed by edges which are explicitly signed by the signer. A property of the transitive signature scheme enables such scenario is called composability. Composability means that by knowing signatures on t...

In this paper, we prove that every vertex-transitive graph can be expressed as the edge-disjoint union of symmetric graphs. We define a multicycle graph and conjecture that every vertex-transitive graph cam be expressed as the edge-disjoint union of multicycles. We verify this conjecture for several subclasses of vertextransitive graphs, including Cayley graphs, multidimensional circulants, and...

It is shown that the graph QQ) C obtained from the 7 -cube QQ) by deletion of a perfect Hamming code C has a spanning self-complementary subgraph which is edge-transitive but not vertextransitive and also extremal among all the cube subgraphs which are square-blocking and codeavoiding. Our work uses combinatorial techniques involving orientations on the Fano plane and the resulting Steiner trip...

We introduce some constructions of weakly distance-regular digraphs of girth 2, and prove that a certain quotient digraph of a commutative weakly distance-transitive digraph of girth 2 is a distancetransitive graph. As an application of the result, we obtain some examples of weakly distanceregular digraphs which are not weakly distance-transitive. Moreover, a class of commutative weakly distanc...