Every finite, self-dual, regular (or chiral) 4-polytope of type {3, q, 3} has a trivalent 3-transitive (or 2-transitive) medial layer graph. Here, by dropping self-duality, we obtain a construction for semisymmetric trivalent graphs (which are edgebut not vertex-transitive). In particular, the Gray graph arises as the medial layer graph of a certain universal locally toroidal regular 4-polytope.

Since decades transitive graphs are a topic of great interest. The study of s-edge transitive (undirected) graphs goes back to Tutte [13], who showed that finite cubic graphs cannot be s-edge transitive for s> 5. Weiss [14] proved several years later that the only finite connected s-edge transitive graphs with s = 8 are the cycles. Considering directed graphs the situation is much more involved...

The results of J. Širáň and the first author (Australasian J. Combin. 10 (1994)) are generalized, and new formulas for the number of closed walks of length p or pq, where p and q are primes, valid for all vertex-transitive graphs are found. Based on these formulas, several simple tests for vertex-transitivity are presented, as well as lower bounds on the orders of the smallest vertexand arc-tra...

Let φ be Euler’s phi function. We prove that a vertex-transitive graph 0 of order n, with gcd(n, φ(n)) = 1, is isomorphic to a circulant graph of order n if and only if Aut(0) contains a transitive solvable subgroup. As a corollary, we prove that every vertex-transitive graph 0 of order n is isomorphic to a circulant graph of order n if and only if for every such 0, Aut(0) contains a transitive...

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...

A con guration is weakly ag transitive if its group of automor phisms acts intransitively on ags but the group of all automorphisms and anti automorphisms acts transitively on ags It is shown that weakly ag transitive con gurations are in one to one correspondence with bipartite  arc transitive graphs of girth not less than Sev eral in nite families of weakly ag transitive con gurations are gi...

A binary relation R on a set S is transitive iff for all a, b, c ∈ S, if aRb and bRc, then aRc. This almost never applies to the relations logicians tend to think of as consequence relations; where such relations are relations on a set at all, they are rarely transitive. Yet it is common to hear consequence relations described as ‘transitive’, and to see rules imposed to ensure ‘transitivity’ o...