Constructing even radius tightly attached half-arc-transitive graphs of valency four
Journal of Algebraic Combinatorics: An International Journal
Unexpected symmetries in unstable graphs
Journal of Combinatorial Theory Series B
A classification of tightly attached half-arc-transitive graphs of valency 4
Journal of Combinatorial Theory Series B
On quartic half-arc-transitive metacirculants
Journal of Algebraic Combinatorics: An International Journal
An infinite family of half-arc-transitive graphs with universal reachability relation
European Journal of Combinatorics
On the vertex-stabiliser in arc-transitive digraphs
Journal of Combinatorial Theory Series B
Four Constructions of Highly Symmetric Tetravalent Graphs
Journal of Graph Theory
Tetravalent arc-transitive locally-Klein graphs with long consistent cycles
European Journal of Combinatorics
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In this paper, we first consider graphs allowing symmetry groups which act transitively on edges but not on darts (directed edges). We see that there are two ways in which this can happen and we introduce the terms bi-transitive and semi-transitive to describe them. We examine the elementary implications of each condition and consider families of examples; primary among these are the semi-transitive spider-graphs PS(k,N;r) and MPS(k,N;r). We show how a product operation can be used to produce larger graphs of each type from smaller ones. We introduce the alternet of a directed graph. This links the two conditions, for each alternet of a semi-transitive graph (if it has more than one) is a bi-transitive graph. We show how the alternets can be used to understand the structure of a semi-transitive graph, and that the action of the group on the set of alternets can be an interesting structure in its own right. We use alternets to define the attachment number of the graph, and the important special cases of tightly attached and loosely attached graphs. In the case of tightly attached graphs, we show an addressing scheme to describe the graph with coordinates. Finally, we use the addressing scheme to complete the classification of tightly attached semi-transitive graphs of degree 4 begun by Maru&sbreve;i&cbreve; and Praeger. This classification shows that nearly all such graphs are spider-graphs. © 2003 Wiley Periodicals, Inc. J Graph Theory 45: 1–27, 2004