Let D be a connected balanced digraph. The classical distance dijD from vertex i to j is the length of shortest directed path in D. L Laplacian matrix and L†=(lij†) Moore–Penrose inverse L. resistance rijD then defined by rijD≔lii†+ljj†−2lij†. C collection digraphs, each member which finite union form D1∪D2∪....∪Dk where Dt digraph with Dt∩(D1∪D2∪⋯∪Dt−1) being single vertex, for all 1<t≤k. In t...

A classical result by Erdős and Pósa[3] states that there is a function f : N → N such that for every k, every graph G contains k pairwise vertex disjoint cycles or a set T of at most f(k) vertices such that G− T is acyclic. The generalisation of this result to directed graphs is known as Younger’s conjecture and was proved by Reed, Robertson, Seymour and Thomas in 1996. This so-called Erdős-Pó...

In this paper, we completely determine the connectivity of every infinite circulant digraphs and prove that almost all infinite circulant digraphs are infinitely strongly connected and therefore have both oneand two-way infinite Hamiltonian paths.

Models of complex information processing based on artificial neural networks frequently apply fully connected or random graph structures. However, it is well known that biological neural systems operate on sparsely connected networks having properties quite distinct to random graphs. In this paper, a simple method is introduced for the deterministic generation of strongly connected digraphs. Th...

Mader (1985) proved that every strongly $k$-connected $n$-vertex digraph contains a strongly $k$-connected spanning subgraph with at most $2kn - 2k^2$ edges, and this is sharp. For dense strongly $k$-connected digraphs, the bound can be significantly improved for dense strongly $k$-connected digraphs. Let $\overline{\Delta}(D)$ be the maximum degree of the complement of the underlying undirecte...

A maximally connected digraph is said to be superconnected if every minimum disconnecting set F of vertices is trivial, i.e., it consists of the vertices adjacent to or from a given vertex not belonging to F . In this work, it is shown that a digraph is maximally connected provided that a certain condition on the diameter holds, and superconnected under a more restrictive constraint. These cond...

A digraph is said to be super-connected if every minimum vertex cut is the out-neighbor set or in-neighbor set of a vertex. A digraph is said to be reducible, if there are two vertices with the same out-neighbor set or the same in-neighbor set. In this paper, we prove that a strongly connected arc-transitive oriented graph is either reducible or super-connected. Furthermore, if this digraph is ...

Preface Graphs are often used as a model for networks, for example transport networks, telecommunication systems, a network of servers and so on. Typically, the networks and thus the representing graphs are connected; that means, that there exists a path between all pairs of two vertices in the graph. But sometimes an element of the network fails, for the graphs that means the removal of a vert...