Following Alspach and Parsons, a metacirculant graph is a graph admitting a transitive group generated by two automorphisms ρ and σ , where ρ is (m,n)semiregular for some integers m ≥ 1, n ≥ 2, and where σ normalizes ρ, cyclically permuting the orbits of ρ in such a way that σ has at least one fixed vertex. A halfarc-transitive graph is a vertexand edgebut not arc-transitive graph. In this arti...

A transitive orientation of an undirected graph is an assignment of directions to its edges so that these directed edges represent a transitive relation between the vertices of the graph. Not every graph has a transitive orientation, but every graph can be turned into a graph that has a transitive orientation, by adding edges. We study the problem of adding an inclusion minimal set of edges to ...

Several authors have studied methods to construct the transitive reduction of a directed graph, but little work has been done on how to maintain it. We are motivated by a real-world application which uses a transitively reduced graph at its core and must maintain the transitive reduction over a sequence of graph operations. This paper presents an efficient method to maintain the transitive redu...

In this paper we introduce and study a type of Cayley graph – subnormal graph. We prove that 2-arc transitive is normal or cover complete bipartite $\mathbf{K}_{p^d,p^d}$ with $p$ prime. Then obtain generic method for constructing half-symmetric (namely edge but not arc transitive) graphs.

A regular graph is called semisymmetric if it is edge-transitive but not vertex-transitive. It is proved that the Gray graph is the only cubic semisymmetric graph of order 2p3, where p 3 is a prime.

A regular graph is called semisymmetric if it is edge transitive but not vertex transitive It is proved that the Gray graph is the only cubic semisymmetric graph of order p where p is a prime

The relative fixity of a permutation group is the maximum proportion points fixed by non-trivial element group, and graph its automorphism viewed as on vertex-set graph. We prove in this paper that connected 2-arc-transitive graphs valence tends to 0 number vertices grows infinity. same result for class arc-transitive prime valence, more generally, any locally-L graphs, where L quasiprimitive g...

In this paper, a complete classification of finite simple cubic vertex-transitive graphs girth 6 is obtained. It proved that every such graph, with the exception Desargues graph on 20 vertices, either skeleton hexagonal tiling torus, truncation an arc-transitive triangulation closed hyperbolic surface, or 6-regular respect to dihedral scheme. Cubic larger than are also discussed.

Let Γ be a finite G-symmetric graph whose vertex set admits a non-trivial Ginvariant partition B with block size v. A framework for studying such graphs Γ was developed by Gardiner and Praeger which involved an analysis of the quotient graph ΓB relative to B, the bipartite subgraph Γ[B,C] of Γ induced by adjacent blocks B,C of ΓB and a certain 1-design D(B) induced by a block B ∈ B. The present...

We consider economical representations for the path information in a directed graph. A directed graph G is said to be a transitive reduction of the directed graph G provided that (i) G has a directed path from vertex u to vertex v if and only if G has a directed path from vertex u to vertex v, and (ii) there is no graph with fewer arcs than G satisfying condition (i). Though directed graphs wit...