If x is a vertex of a digraph D, then we denote by d(x) and d−(x) the outdegree and the indegree of x, respectively. The global irregularity of a digraph D is defined by ig(D) = max{d+(x), d−(x)}−min{d+(y), d−(y)} over all vertices x and y of D (including x = y). If ig(D) = 0, then D is regular and if ig(D) ≤ 1, then D is called almost regular. Recently, L. Volkmann and S. Winzen showed that ev...

A digraph D is a local out-tournament if the outset of every vertex: is a tournament. Here, we use local out-tournaments, whose strong components are upset tournaments, to explore the corresponding ranks of the adjacency matrices. Of specific interest is the out-tournament whose adjacency matrix has boolean, nonnegative integer, term, and real rank all equal to the number of vertices, n. Corres...

We propose an O(min{m+n logn,m log∗n}) to find a minmax strongly connected spanning subgraph of a digraph with n nodes and m arcs. A generalization of this problem called the minmax strongly connected subgraph problem with node penalties is also considered. An O(m logn) algorithm is proposed to solve this general problem. We also discuss ways to improve the average complexity of this algorithm.

The following (solitaire) game is considered: Initially each node of a simple, connected , nite graph contains a nite number of chips. A move consists in ring all nodes with at least as many chips as their degree, where ring a node corresponds to sending one of the node's chips to each one of the node's neighbors. In Bi89] it was conjectured that every parallel chip ring game played on an N nod...

where the minimum is taken over all subsets X of E(G) such that ω(G − X) − c > 0. In this paper, we establish a relationship 7 between λc(G) and τc−1(G), which gives a characterization of the edge-connectivity of a graph G in terms of the spanning tree 8 packing number of subgraphs of G. The digraph analogue is also obtained. The main results are applied to show that if a graph G is 9 s-hamilto...

For a given $2$-partition $(V_1,V_2)$ of the vertices of a (di)graph $G$, we study properties of the spanning bipartite subdigraph $B_G(V_1,V_2)$ of $G$ induced by those arcs/edges that have one end in each $V_i$. We determine, for all pairs of non-negative integers $k_1,k_2$, the complexity of deciding whether $G$ has a 2-partition $(V_1,V_2)$ such that each vertex in $V_i$ has at least $k_i$ ...

In a directed graph, a star is an arborescence with at least one arc, in which the root dominates all the other vertices. A galaxy is a vertex-disjoint union of stars. In this paper, we consider the Spanning Galaxy Problem of deciding whether a digraph D has a spanning galaxy or not. We show that although this problem is NP-complete (even when restricted to acyclic digraphs), it becomes polynom...

In this paper we propose a new decentralized algorithm to solve the consensus on the average problem on arbitrary strongly connected digraphs through a gossip algorithm based on broadcasts. We directly extend previous results by not requiring that the digraph is balanced. Our algorithm is an improvement respect to known gossip algorithms based on broadcasts in that the average of the initial st...

We are interested in the construction of the largest possible vertex symmetric digraphs with the property that between any two vertices there is a walk of length two (that is, they are 2-reachable). Other than a theoretical interest, these digraphs and their generalizations may be used as models of interconnection networks for implementing parallelism. In these systems many nodes are connected ...