A digraph (D) is symmetric if (u,v) an arc of D and (v,u) also D. If a primitive contains d loops, then it said to be with loops. The m-competition index (generalized competition index) extension the exponent scrambling index. has been applied memoryless communication systems in recent years. In this article, we assume that Sn(d) represents set all digraphs n vertices where 1≤d≤n. We study indi...

We show that the independent spanning tree conjecture on digraphs is true if we restrict ourselves to line digraphs. Also, we construct independent spanning trees with small depths in iterated line digraphs. From the results, we can obtain independent spanning trees with small depths in de Bruijn and Kautz digraphs that improve the previously known upper bounds on the depths.

Eilers et al. have recently completed the geometric classification of unital graph $C^\ast$-algebras up to Morita equivalence using a set moves on corresponding digraphs. We explore question whether these preserve nonzero elements spectrum finite digraph, which in this paper is allowed loops and parallel edges. consider several different digraph spectra that been studied literature, answering f...

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...

The Moore bound for a diregular digraph of degree d and diameter k is M d;k = 1 + d + : : : + d k. It is known that digraphs of order M d;k do not exist for d > 1 and k > 1. In this paper we study digraphs of order M d;k ? 1, that is, digraphs with defect 1, denoted by (d; k)-digraphs. If G is a (d; k)-digraph, then for each vertex v of G there exists a vertex w (called the repeat of v) such th...

Kautz digraphs K(d, `) are a well-known family of dense digraphs, widely studied as a good model for interconnection networks. Closely related with these, the cyclic Kautz CK(d, `) and the subKautz sK(d, 2) digraphs were recently introduced by Böhmová, Huemer and the author. In this paper we propose a new method to obtain the complete spectra of subKautz sK(d, 2) and cyclic Kautz CK(d, 3) digra...

We consider a directed version of Deza graphs. A digraph is said to be a Deza digraph if it is regular and the number of common out-neighbors of any two distinct vertices takes on at most two values. We introduce some constructions and develop some basic theory. In particular, constructions from Cayley digraphs, graph lexicograph products and graph directed products are described. Moreover, we ...

We study the list homomorphism and retraction problems for the class of reflexive digraphs (digraphs in which each vertex has a loop). These problems have been intensively studied in the case of undirected graphs, but the situation seems more complex for digraphs. We also focus on an intermediate ‘subretraction’ problem. It turns out that the complexity of the subretraction problem can be class...

A digraph D is k-transitive if the existence of a directed path (v0, v1, . . . , vk), of length k implies that (v0, vk) ∈ A(D). Clearly, a 2-transitive digraph is a transitive digraph in the usual sense. Transitive digraphs have been characterized as compositions of complete digraphs on an acyclic transitive digraph. Also, strong 3 and 4-transitive digraphs have been characterized. In this work...