We derive sharp lower bounds for the first and the second Zagreb indices (M1 and M2 respectively) for trees and chemical trees with the given number of pendent vertices and find optimal trees. M1 is minimized by a tree with all internal vertices having degree 4, while M2 is minimized by a tree where each “stem” vertex is incident to 3 or 4 pendent vertices and one internal vertex, while the res...

The graphs having the maximum value of certain bond incident degree indices (including second Zagreb index, general sum-connectivity and zeroth-order Randi? index) in class all connected with fixed order number pendent vertices are characterized this paper. problem finding minimum values index from aforementioned is also addressed. One obtained results about first has already been proved papers...

We analyse a randomly growing graph model in which the average degree is asymptotically equal to constant times square root of number vertices, and clustering coefficient rather small. In every step, we choose two vertices uniformly at random, check whether they are connected or not, either add new edge delete one vertex graph. This dependence on status connection chosen makes total random afte...

In the classical Erdős–Rényi model of random graphs, when the number of edges is proportional to the number of vertices, the degree distribution is approximately Poisson with a tail decreasing even faster than exponentially. However, in many real life networks power law degree distributions were observed with different exponents. To introduce a more realistic model for the evolution of random n...

The inflation $G_{I}$ of a graph $G$ with $n(G)$ vertices and $m(G)$ edges is obtained from $G$ by replacing every vertex of degree $d$ of $G$ by a clique, which is isomorph to the complete graph $K_{d}$, and each edge $(x_{i},x_{j})$ of $G$ is replaced by an edge $(u,v)$ in such a way that $uin X_{i}$, $vin X_{j}$, and two different edges of $G$ are replaced by non-adjacent edges of $G_{I}$. T...

We study a dynamically evolving random graph which adds vertices and edges using preferential attachment and deletes vertices randomly. At time t, with probability α1 > 0 we add a new vertex ut and m random edges incident with ut. The neighbours of ut are chosen with probability proportional to degree. With probability α − α1 ≥ 0 we add m random edges to existing vertices where the endpoints ar...

Consider an arbitrary 3-regular graph G = (V,E) with connectivity 1. By the definition it has a cut vertex v ∈ V , and since its degree is three, there are either 2 or 3 connected components in G \ v. Moreover, in every connected component of G \ v there exist either one or two vertices with degree 2, and all others have degree 3. On the other hand, there has to be an even number of vertices of...

Many real-world networks were found to be highly clustered, and contain a large amount of small cliques. We here investigate the number cliques any size k contained in geometric inhomogeneous random graph: scale-free network model containing geometry. The interplay between scale-freeness geometry ensures that connections are likely form either high-degree vertices, or close by vertices. At same...

let $g$ be a finite group and $pi(g)$ be the set of all the prime‎ ‎divisors of $|g|$‎. ‎the prime graph of $g$ is a simple graph‎ ‎$gamma(g)$ whose vertex set is $pi(g)$ and two distinct vertices‎ ‎$p$ and $q$ are joined by an edge if and only if $g$ has an‎ ‎element of order $pq$‎, ‎and in this case we will write $psim q$‎. ‎the degree of $p$ is the number of vertices adjacent to $p$ and is‎ ...

Let $G$ be a weighted digraph, $s$ and $t$ be two vertices of $G$, and $t$ is reachable from $s$. The logical $s$-$t$ min-cut (LSTMC) problem states how $t$ can be made unreachable from $s$ by removal of some edges of $G$ where (a) the sum of weights of the removed edges is minimum and (b) all outgoing edges of any vertex of $G$ cannot be removed together. If we ignore the second constraint, ca...