In an in nite digraph D an edge e is reachable from an edge e if there exists an alternating walk in D whose initial and terminal edges are e and e Reachability is an equivalence relation and if D is arc transitive then this relation is either universal or all of its equiv alence classes induce isomorphic bipartite digraphs In Question Cameron Praeger and Wormald asked if there exist highly arc...

In a non-complete graph $Gamma$, a vertex triple $(u,v,w)$ with $v$ adjacent to both $u$ and $w$ is called a $2$-geodesic if $uneq w$ and $u,w$ are not adjacent. The graph $Gamma$ is said to be   $2$-geodesic transitive if its automorphism group is transitive on arcs, and also on 2-geodesics. We first produce a reduction theorem for the family of $2$-geodesic transitive graphs of prime power or...

We show how to exploit symmetries of a graph to efficiently compute the fastest mixing Markov chain on the graph (choose the transition probabilities on the edges to minimize the second-largest eigenvalue modulus of the transition probability matrix). Exploiting symmetry can lead to significant reduction in both number of optimization variables and size of matrices in solving the corresponding ...

A transitive signature scheme allows a signer to publish a graph in an authenticated and cost-saving manner. The resulting authenticated graph is indeed the transitive closure of the graph constructed by edges which are explicitly signed by the signer. A property of the transitive signature scheme enables such scenario is called composability which means that by knowing signatures on two edges ...

We investigate suborbital graphs for an imprimitive action of the Atkin–Lehner group on a maximal subset of extended rational numbers on which a transitive action is also satisfied. Obtaining edge and some circuit conditions, we examine some combinatorial properties of these graphs.

The notions of (transitive) soft d-algebras, soft edge d-algebras, soft d-algebras, soft dideals, soft d-ideals, soft d-ideals, and d-idealistic (d-idealistic, or d-idealistic) soft dalgebras are introduced. Also, their related properties are surveyed. © 2008 Elsevier Ltd. All rights reserved.