A generalization of some of the Folkman's constructions [13] of the so called semisymmetric graphs, that is regular graphs which are edgebut not vertex-transitive, was given in [22] together with a natural connection of graphs admitting 12 -arc-transitive group actions and certain graphs admitting semisymmetric group actions. This connection is studied in more detail in this paper. In Theorem 2...

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

Abstract If ${\mathbf v} \in {\mathbb R}^{V(X)}$ is an eigenvector for eigenvalue $\lambda $ of a graph X and $\alpha automorphism , then ({\mathbf v})$ also . Thus, it rather exceptional vertex-transitive to have multiplicity one. We study cubic graphs with nontrivial simple eigenvalue, discover remarkable connections arc-transitivity, regular maps, number theory.

Denote by G the set of triples (Γ, X,B), where Γ is a finite X-symmetric graph of valency val(Γ) ≥ 1, B is a nontrivial X-invariant partition of V (Γ) such that ΓB, the quotient graph of Γ with respect to B, is nonempty and Γ is not a multicover of ΓB. In this article, for any given X-symmetric graph Σ, we aim to give a sufficient and necessary condition for the existence of (Γ, X,B) ∈ G, such ...

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 ...

In this note we consider two related infinite families of graphs, which generalize the Petersen and the Coxeter graph. The main result proves that these graphs are cores. It is determined which of these graphs are vertex/edge/arc-transitive or distance-regular. Girths and odd girths are computed. A problem on hamiltonicity is posed.

In this paper, D = (V (D), A(D)) denotes a loopless directed graph (digraph) with at most one arc from u to v for every pair of vertices u and v of V (D). Given a digraph D, we say that D is 3-quasi-transitive if, whenever u → v → w → z in D, then u and z are adjacent. In [3], Bang-Jensen introduced 3-quasi-transitive digraphs and claimed that the only strong 3quasi-transitive digraphs are the ...

We give a unified approach to analysing, for each positive integer s, a class of finite connected graphs that contains all the distance transitive graphs as well as the locally s-arc transitive graphs of diameter at least s. A graph is in the class if it is connected and if, for each vertex v, the subgroup of automorphisms fixing v acts transitively on the set of vertices at distance i from v, ...

Two problems of Cameron, Praeger, and Wormald [Infinite highly arc transitive digraphs and universal covering digraphs, Combinatorica (1993)] are resolved. First, locally finite highly arc-transitive digraphs with universal reachability relation are presented. Second, constructions of two-ended highly arc-transitive digraphs are provided, where each ‘building block’ is a finite bipartite digrap...