A vertex-deleted unlabeled subdigraph of a digraph D is a card of D. A dacard specifies the degree triple (a, b, c) of the deleted vertex along with the card, where a and b are respectively the indegree and outdegree of v and c is the number of symmetric pairs of arcs (each pair considered as unordered edge) incident with v. The degree (triple) associated reconstruction number, drn(D), of a dig...

Hamidoune’s connectivity results [11] for hierarchical Cayley digraphs are extended to Cayley coset digraphs and thus to arbitrary vertex transitive digraphs. It is shown that if a Cayley coset digraph can be hierarchically decomposed in a certain way, then it is optimally vertex connected. The results are obtained by extending the methods used in [11]. They are used to show that cycle-prefix g...

A simple decomposition for graphs yields generating functions for counting graphs by edges and connected components. A change of variables gives a new interpretation to the Tutte polynomial of the complete graph involving inversions of trees. The relation between the Tutte polynomial of the complete graph and the inversion enumerator for trees is generalized to the Tutte polynomial of an arbitr...

Generalizing the idea of viewing a digraph as a model of a linear map, we suggest a multi-variable analogue of a digraph, called a hydra, as a model of a multi-linear map. Walks in digraphs correspond to usual matrix multiplication while walks in hydras correspond to the tensor multiplication introduced by Robert Grone in 1987. By viewing matrix multiplication as a special case of this tensor m...

Our aim in this note is to prove a conjecture of Bondy, extending a classical theorem of Dirac to edge-weighted digraphs: if every vertex has out-weight at least 1 then the digraph contains a path of weight at least 1. We also give several related conjectures and results concerning heavy cycles in edge-weighted digraphs.

Let D be an edge-coloured digraph, V (D) will denote the set of vertices of D; a set N ⊆ V (D) is said to be a kernel by monochromatic paths of D if it satisfies the following two conditions: For every pair of different vertices u, v ∈ N there is no monochromatic directed path between them and; for every vertex x ∈ V (D) − N there is a vertex y ∈ N such that there is an xy -monochromatic direct...

The Klee-Quaife problem is finding the minimum order μ(d, c, v) of the (d, c, v) graph, which is a c-vertex connected v-regular graph with diameter d. Many authors contributed finding μ(d, c, v) and they also enumerated and classified the graphs in several cases. This problem is naturally extended to the case of digraphs. So we are interested in the extended Klee-Quaife problem. In this paper, ...