The class of connected 3-colored digraphs containing exactly one nonsingular cycle is considered in this article. The main objective is to study the smallest Laplacian eigenvalue and the corresponding eigenvectors of such graphs. It is shown that the smallest Laplacian eigenvalue of such a graph can be realized as the algebraic connectivity (second smallest Laplacian eigenvalue) of a suitable u...

The sizes of the cycles and unicyclic components in the random graph G(n, n/2 ± s), where n ≪ s ≪ n, are studied using the language of point processes. This refines several earlier results by different authors. Asymptotic distributions of various random variables are given; these distributions include the gamma distributions with parameters 1/4, 1/2 and 3/4, as well as the Poisson–Dirichlet and...

In this note, we show how the determinant of the q-distance matrix Dq(T ) of a weighted directed graph G can be expressed in terms of the corresponding determinants for the blocks of G, and thus generalize the results obtained by Graham et al. [R.L. Graham, A.J. Hoffman and H. Hosoya, On the distance matrix of a directed graph, J. Graph Theory 1 (1977) 85–88]. Further, by means of the result, w...

let $g=(v‎, ‎e)$ be a graph with $p$ vertices and $q$ edges‎. ‎an emph{acyclic‎ ‎graphoidal cover} of $g$ is a collection $psi$ of paths in $g$‎ ‎which are internally-disjoint and cover each edge of the graph‎ ‎exactly once‎. ‎let $f‎: ‎vrightarrow {1‎, ‎2‎, ‎ldots‎, ‎p}$ be a bijective‎ ‎labeling of the vertices of $g$‎. ‎let $uparrow!g_f$ be the‎ ‎directed graph obtained by orienting the...

The Erdős-Rényi process begins with an empty graph on n vertices, with edges added randomly one at a time to the graph. A classical result of Erdős and Rényi states that the Erdős-Rényi process undergoes a phase transition, which takes place when the number of edges reaches n/2 (we say at time 1) and a giant component emerges. Since this seminal work of Erdős and Rényi, various random graph mod...

In this note, we study the unicyclic representation introduced in O’Cinneide[21]. First, we present a counterexample to the conjecture that every PH -representation has an equivalent unicyclic representation of the same order. Then we show that the conjecture holds if the order of the PH -representation is 3. We also introduce an algorithm for computing a unicyclic generator of order 3, which P...

The transmission of a vertex v graph G is the sum distances from to all other vertices in . A irregular if its have pairwise different transmissions. starlike tree T ( k 1 , … t ) obtained by attaching an isolated pendant paths lengths respectively. It proved that + ? 2 odd order, then it irregular. ? 3 and only ? { r : } Additional infinite families among trees bi-starlike are determined. Tran...

Let G = (V,E) be a graph. A set S ⊂ V (G) is a hop dominating set of G if for every v ∈ V − S, there exists u ∈ S such that d(u, v) = 2. The minimum cardinality of a hop dominating set of G is called a hop domination number of G and is denoted by γh(G). In this paper we characterize the family of trees and unicyclic graphs for which γh(G) = γt(G) and γh(G) = γc(G) where γt(G) and γc(G) are the ...

Let $D$ be a weighted oriented graph and $I(D)$ its edge ideal. If contains an induced odd cycle of length $2n+1$, under certain condition we show that $ {I(D)}^{(n+1)} \neq {I(D)}^{n+1}$. We give necessary sufficient for the equality ordinary symbolic powers ideal having each in some it. characterize naturally unicyclic graphs with unique cycles even their ideals. D^{\prime} obtained from afte...

A cactus graph is a connected graph in which any two cycles have at most one vertex in common. Let γ(G) and γc(G) be the domination number and connected domination number of a graph G, respectively. We can see that γ(G) ≤ γc(G) for any graph G. S. Arumugam and J. Paulraj Joseph [1] have characterized trees, unicyclic graphs and cubic graphs with equal domination and connected domination numbers...