A connected graph with edge connectivity λ and minimum degree δ is called maximally edge-connected if λ = δ. A strongly connected digraph with arc connectivity λ and minimum degree δ is called maximally arc-connected if λ = δ. In this paper, some new sufficient conditions are presented for maximally edge-connected graphs and arc-connected digraphs. We only consider finite graphs (digraphs) with...

Biological neural systems and the majority of other realworld networks have topologies significant different from fully or randomly connected structures, which are frequently applied for the definition of artificial neural networks (ANN). In this work we introduce a deterministic process generating strongly connected directed graphs of fractal dimension having connectivity structures very disti...

A group G of permutations of a set is primitive if it acts transitively on , and the only G-invariant equivalence relations on are the trivial and universal relations. A digraph is primitive if its automorphism group acts primitively on its vertex set, and is infinite if its vertex set is infinite. It has connectivity one if it is connected and there exists a vertex α of , such that the induced...

Wada, K. and K. Kawaguchi, Efficient fault-tolerant fixed routings on (k+ 1)connected digraphs, Discrete Applied Mathematics 37/38 (1992) 539-552. Consider a directed communication network G in which a limited number of directed link and/or node faults F might occur. A routing Q for the network (a fixed path for each ordered pair of nodes) must be chosen without knowing which components might b...

Given a directed graph G = (V;E), a natural problem is to choose a minimum number of the edges in E such that, for any two vertices u and v, if there is a path from u to v in E, then there is a path from u to v among the chosen edges. We show that in graphs having no directed cycle with more than three edges, this problem is equivalent to Maximum Bipartite Matching. This leads to a small improv...

Let −→ G be a digraph and A( −→ G) be the adjacency matrix of −→ G . Let D( −→ G) be the diagonal matrix with outdegrees of vertices of −→ G and Q( −→ G) = D( −→ G) + A( −→ G) be the signless Laplacian matrix of −→ G . The spectral radius of Q( −→ G) is called the signless Laplacian spectral radius of −→ G . In this paper, we determine the unique digraph which attains the maximum (or minimum) s...

For a nonnegative n× n matrix A, we find that there is a polynomial f(x) ∈ R[x] such that f(A) is a positive matrix of rank one if and only if A is irreducible. Furthermore, we show that the lowest degree such polynomial f(x) with tr f(A) = n is unique. Thus, generalizing the well-known definition of the Hoffman polynomial of a strongly connected regular digraph, for any irreducible nonnegative...

A linear recurrence relation is derived for the number of unlabelled initially connected acyclic automata. The coefficients of this relation are determined by another, alternating, recurrence relation. The latter determines, in particular, the number of acyclic automata with labelled states. Certain simple enumerative techniques developed by the author for counting initially connected automata ...

Let D be a digraph on [Formula: see text]. Then the sequence [Formula: see text] is called the degree sequence of D. For any given sequence of pairs of integers [Formula: see text], if there exists a k-arc strongly connected digraph D such that d is the degree sequence of D, then d is realizable and D is a realization of d. In this paper, characterizations for k-arc-connected realizable sequenc...