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...

An automorphism of a graph is called quasi-semiregular if it fixes unique vertex the and its remaining cycles have same length. This kind symmetry graphs was first investigated by Kutnar, Malnič, Martínez Marušič in 2013, as generalization well-known problem regarding existence semiregular automorphisms vertex-transitive graphs. Symmetric valency three or four, admitting automorphism, been clas...

Recently, Hua et al. defined a new topological index based on degrees and inverse of distances between all pairs of vertices. They named this new graph invariant as reciprocal degree distance as 1 { , } ( ) ( ( ) ( ))[ ( , )] RDD(G) = u v V G d u  d v d u v , where the d(u,v) denotes the distance between vertices u and v. In this paper, we compute this topological index for Grassmann graphs.

A subgroup G of automorphisms of a graph X is said to be 2 -arc-transitive if it is vertexand edgebut not arc-transitive. The graph X is said to be 2 -arc-transitive if Aut X is 1 2 -arc-transitive. The interplay of two different concepts, maps and hypermaps on one side and 2 -arc-transitive group actions on graphs on the other, is investigated. The correspondence between regular maps and 2 -ar...

A graph G is k-ordered if for any sequence of k distinct vertices v1, v2, . . . , vk of G there exists a cycle in G containing these k vertices in the specified order. In 1997, Ng and Schultz posed the question of the existence of 4-ordered 3-regular graphs other than the complete graph K4 and the complete bipartite graph K3,3. In 2008, Meszaros solved the question by proving that the Petersen ...

An automorphism of a 7nite simple graph is an adjacency automorphism if for every vertex x∈V ( ), either x = x or x is adjacent to x in . An adjacency automorphism 7xing no vertices is a shift. A connected graph is strongly adjacency-transitive (respectively, uniquely shift-transitive) if there is, for every pair of adjacent vertices x; y∈V ( ), an adjacency automorphism (respectively, a unique...

We extend the main result of R. Jajcay and J. Širáň [Australasian J. Combin. 10 (1994), 105–114] to produce new classes of vertex-transitive nonCayley graphs.

We prove that, given a finite graph ? satisfying some mild conditions, there exist infinitely many tetravalent half-arc-transitive normal covers of ?. Applying this result, we establish the existence infinite families graphs with certain vertex stabilizers, and classify stabilizers up to order 28 connected graphs. This sheds new light on longstanding problem classifying

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 ...

Transitive permutation groups having a non self paired suborbit of length are investigated via the corresponding orbital graphs If G is such a group and X is the orbital graph associated with a sub orbit of length of G which is not self paired then X has valency and admits a vertex and edge but not arc transitive action of G There is a natural balanced orientation of the edge set of X induced a...