Let U(n, g) and B(n, g) be the set of unicyclic graphs and bicyclic graphs on n vertices with girth g, respectively. Let B1(n, g) be the subclass of B(n, g) consisting of all bicyclic graphs with two edge-disjoint cycles and B2(n, g) = B(n, g)\B1(n, g). This paper determines the unique graph with the maximal signless Laplacian spectral radius among all graphs in U(n, g) and B(n, g), respectivel...

Let H = (V (H), E(H)) be a simple connected graph of order n with the vertex set V (H) and the edge set E(H). We consider a blow-up graph G[H ]. We are interested in the following problem. We have to decide whether there exists a blow-up graph G[H ], with edge densities satisfying special conditions (homogeneous or inhomogeneous), such that the graph H does not appear in a blow-up graph as a tr...

We consider the number of vertex independent sets i(G). In general, the problem of determining the value of i(G) is NP -complete. We present several upper and lower bounds for i(G) in terms of order, size or independence number. We obtain improved bounds for i(G) on restricted graph classes such as the bipartite graphs, unicyclic graphs, regular graphs and claw-free graphs.

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...