Distance-transitive and distance-regular digraphs

Weakly Distance-regular Digraphs

Some Notes on Distance-Transitive and Distance-Regular Graphs

These are notes from lectures given in the Queen Mary Combinatorics Study 1 Introductory Definitions In these notes, Γ = (V Γ, EΓ) denotes a graph, as we will use G to denote a group. All graphs considered will be simple, finite, connected and undirected. Definition An automorphism of Γ is a bijective function g : V Γ → V Γ such that v ∼ w if and only if g(v) ∼ g(w). The set of all automorphism...

Weakly Distance-Regular Digraphs of Girth 2

In this paper, we give two constructions of weakly distance-regular digraphs of girth 2, and prove that certain quotient digraph of a commutative weakly distancetransitive digraph of girth 2 is a distance-transitive graph. As an application of the result, we not only give some constructions of weakly distance-regular digraphs which are not weakly distance-transitive, but determine a special cla...

Minimum Cuts of Distance-Regular Digraphs

In this paper, we investigate the structure of minimum vertex and edge cuts of distance-regular digraphs. We show that each distance-regular digraph Γ, different from an undirected cycle, is super edge-connected, that is, any minimum edge cut of Γ is the set of all edges going into (or coming out of) a single vertex. Moreover, we will show that except for undirected cycles, any distance regular...

Distance Sequences In Locally Infinite Vertex-Transitive Digraphs

We prove that the out-distance sequence {f(k)} of a vertex-transitive digraph of finite or infinite degree satisfies f(k + 1) ≤ f(k) for k ≥ 1, where f(k) denotes the number of vertices at directed distance k from a given vertex. As a corollary, we prove that for a connected vertextransitive undirected graph of infinite degree d, we have f(k) = d for all k, 1 ≤ k < diam(G). This answers a quest...