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 digraph H is infused in a digraph G if the vertices of H are mapped to vertices of G (not necessarily distinct), and the edges of H are mapped to edge-disjoint directed paths of G joining the corresponding pairs of vertices of G. The algorithmic problem of determining whether a fixed graph H can be infused in an input graph G is polynomial-time solvable for all graphs H (using paths instead o...

A digraph G = (V, E) with diameter D is said to be s-geodetic, for 1 ≤ s ≤ D, if between any pair of (not necessarily different) vertices x, y ∈ V there is at most one x → y path of length ≤ s. Thus, any loopless digraph is at least 1-geodetic. A similar definition applies for a graph G, but in this case the concept is closely related to its girth g, for then G is s-geodetic with s = b(g − 1)/2...

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...

We show that a digraph which contains a directed 2-factor and has minimum in-degree and out-degree at least four has two non-isomorphic directed 2-factors. As a corollary we deduce that every graph which contains a 2factor and has minimum degree at least eight has two non-isomorphic 2factors. In addition we construct: an infinite family of strongly connected 3-diregular digraphs with the proper...

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.

We introduce a number of problems regarding edge-color modifications in edge-colored graphs and digraphs. Consider a property π, a c-edge-colored graph G not satisfying π, and an edge-recoloring cost matrix R = [rij ]c×c where rij ≥ 0 denotes the cost of changing color i of edge e to color j. Basically, in this kind of problem the idea is to change the colors of one or more edges of G in order ...

Let G be a connected graph with minimum degree id="M3"> δ and vertex-connectivity id="M4"> κ . The id="M5"> is id="M6"> k -connected if id="M7"> ≥ , maximally id="M8"> = super-connected every vertex-cut isol...