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.

Symmetric properties of some molecular graphs on the torus are studied. In particular we determine which cubic cyclic Haar graphs are 1-regular, which is equivalent to saying that their line graphs are 1 2-arc-transitive. Although these symmetries make all vertices and all edges indistinguishable , they imply intrinsic chirality of the corresponding molecular graph.

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 ...

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...

Let D be a locally finite, connected, 1-arc transitive digraph. It is shown that the reachability relation is not universal in D provided that the stabilizer of an edge satisfies certain conditions which seem to be typical for highly arc transitive digraphs. As an implication, the reachability relation cannot be universal in highly arc transitive digraphs with prime inor out-degree. Two differe...

For a given finite connected graph , a group H of automorphisms of and a finite group A, a natural question can be raised as follows: Find all the connected regular coverings of having A as its covering transformation group, on which each automorphism in H can be lifted. In this paper, we investigate the regular coverings with A = Zp , an elementary abelian group and get some new matrix-theoret...

Quite a lot of attention has been paid recently to the construction of edgeor arc-transitive covers of symmetric graphs. In most cases, the approach has involved voltage graph techniques, which are excellent for finding regular covers in which the group of covering transformations is either cyclic or elementary abelian, or more generally, homocyclic, but are not so easy to use when the covering...

Abstract A connected graph of order n admitting a semiregular automorphism / k is called -multicirculant. Highly symmetric multicirculants small valency have been extensively studied, and several classification results exist for cubic vertex- arc-transitive multicirculants. In this paper, we study the broader class vertex-transitive graphs an /3 or larger that may not be semiregular. particular...

a graph $gamma$ is said to be vertex-transitive or edge‎- ‎transitive‎ ‎if the automorphism group of $gamma$ acts transitively on $v(gamma)$ or $e(gamma)$‎, ‎respectively‎. ‎let $gamma=cay(g,s)$ be a cayley graph on $g$ relative to $s$‎. ‎then, $gamma$ is said to be normal edge-transitive‎, ‎if $n_{aut(gamma)}(g)$ acts transitively on edges‎. ‎in this paper‎, ‎the eigenvalues of normal edge-tra...

A cubic (trivalent) graph F is said to be 4-arc-transitive if its automorphism group acts transitively on the 4-arcs of r (where a 4-arc is a sequence «;0, vv...,vi of vertices of F such that t;,_j is adjacent to vt for 1 ^ I < 4, and vt-1 ^ vi+1 for 1 < i < 4). In his investigations into graphs of this sort, Biggs defined a family of groups 4(a), for m = 3 ,4 ,5 . . . , each presented in terms...