A distributed Nash equilibrium seeking algorithm is presented for networked games. We assume an incomplete information available to each player about the other players’ actions. The players communicate over a strongly connected digraph to send/receive the estimates of the other players’ actions to/from the other local players according to a gossip communication protocol. Due to asymmetric infor...

A deterministic finite automaton (dfa) of n states over a k-letter alphabet can be seen as a digraph with n vertices which all have exactly k labeled out-arcs (k-out digraph). In 1973 Grusho [16] first proved that with high probability (whp) in a random k-out digraph there is a strongly connected component (scc) of linear size that is reachable from all vertices, i.e., a giant. He also proved t...

• We give a 1.5-approximation for half-integral weighted MSCSS. Our matches known integrality gap lower bound. 2 − f approximation if LP values are bounded below by . In the minimum strongly connected spanning subgraph ( WMSCSS ) problem we must purchase minimum-cost of digraph. show that linear program (LP) solutions can be efficiently rounded to integral at multiplicative 1.5 cost. This round...

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...

Let D = (V,E) be a digraph and u, v ∈ V. The metric maximum distance is defined by md(u,v) max {d⃗(u,v), d⃗(v,u)}, where d⃗(u,v) denote the length of shortest directed u − path in D. m-eccentricity vertex mecc(v) {md(v,u) : V(D)}. m-center mC(D) strongly connected consists all vertices with minimum m-eccentricity, m-periphery mPer(D) relationship between two digraphs their lexicographic product st...

Given a digraph D, the complementarity spectrum of is defined as set eigenvalues its adjacency matrix. This has been shown to be useful in several fields, particularly spectral graph theory. The differences between properties for (undirected) graphs and digraphs, makes study latter particular interest, characterizing strongly connected digraphs with small number non trivial problem. Recently, o...

Let G be a strongly connected digraph with distance matrix D(G) and let Tr(G) the diagonal vertex transmissions of G. For any real ? [0, 1], define D?(G) as = ?Tr(G) + (1-?)D(G). The D? spectral radius is D?(G). In this paper, we first give some upper lower bounds for characterize extremal digraphs. Moreover, digraphs that are not transmission regular, bound on difference between maximum radius...

The Road Coloring Conjecture is an old and classical conjecture posed in Adler and Weiss (1970); Adler et al. (1977). Let G be a strongly connected digraph with uniform out-degree 2. The Road Coloring Conjecture states that, under a natural (necessary) condition that G is “aperiodic”, the edges of G can be colored red and blue such that “universal driving directions” can be given for each verte...

Given a nite directed graph, a coloring of its edges turns the graph into a nite-state automaton. A k-synchronizing word of a deterministic automaton is a word in the alphabet of colors at its edges that maps the state set of the automaton at least on k-element subset. A coloring of edges of a directed strongly connected nite graph of a uniform outdegree (constant outdegree of any vertex) is k-...

Consider an asymmetric wireless network represented by a digraph G = (V , E). A subset of vertices U is called a strongly connected dominating set (SCDS) if the subgraph induced by U is strongly connected and every vertex not in U has both an in-neighbor in U and an out-neighbor in U . SCDS plays an important role of the virtual backbone in asymmetric wireless networks. Motivated by the constru...