‎In this paper‎, ‎we calculate the expected values of the first and second Zagreb indices‎, ‎denoted as $\textbf{E}\left(M_1\right)$ and $\textbf{E}\left(M_2\right)$ respectively‎, ‎as well as the expected value of the forgotten index‎, ‎$\textbf{E}\left(F\right)$‎, ‎for two models of RANDOM bipartite GRAPHS‎. ‎To evaluate our findings‎, ‎we establish the growth rate by demonstrating that for a RANDOM bipartite graph $G$ of order $n$ in either model‎, ‎the expected value of $M_1(G)$ is $O\left( n^3 \right)$‎. ‎Furthermore‎, ‎we prove that the expected values of $M_2(G)$ and $F(G)$ are both $O\left( n^4 \right)$‎.

We investigate two constructions -the replacement and the zig-zag product of GRAPHS- describing several fascinating connections with Combinatorics, via the notion of expander graph, Group Theory, via the notion of semidirect product and Cayley graph, and with Markov chains, via the Lamplighter RANDOM walk. Many examples are provided.

The result of the research is a proposed model for text analysis and identifying the subject and content of texts on Twitter. In this model, two main phases are implemented for classification. In text mining problems and in text mining tasks in general, because the data used is unstructured text, there is a preprocessing phase to extract the feature from this unstructured data. Done. In the second phase of the proposed method, a multilayer neural network algorithm and RANDOM GRAPHS are used to classify the texts. In fact, this algorithm is a method for classifying a text based on the training model. The results show a significant improvement. Comparing the proposed method with other methods, according to the results, we found that the proposed algorithm has a high percentage of improvement in accuracy and has a better performance than other methods. All the presented statistics and simulation output results of the proposed method are based on the implementation in MATLAB software.

We determine the forbidden induced subGRAPHS for the intersection of the classes of chordal bipartite GRAPHS and line GRAPHS of acyclic directed GRAPHS. This is a first step towards finding the forbidden induced subGRAPHS for the class of line GRAPHS of directed GRAPHS.

A set W ,V (G) is called a resolving set, if for every two distinct vertices u,v 2 V (G) there exists w 2 W such that d(u, w) 6= d(v, w), where d(x,y) is the distance between the vertices x and y. A resolving set for G with minimum cardinality is called a metric basis. A graph with a unique metric basis is called a uniquely dimensional graph. In this paper, we establish a family of graph called Solis graph, and we prove that if G is a minimal edge unique base graph with the base of size two, then G belongs to the Solis GRAPHS family. Finally, an algorithm is given for , nding the metric dimension of a Solis graph.

The Hosoya index $Z(G)$ of a graph $G$ is the total number of matchings in it. In this paper, the recursive formulas of the Hosoya index of semitotal graph $Q(G)$ and total graph $T(G)$ for certain GRAPHS $G$ are obtained. Moreover, we obtain the bounds of the Hosoya index of semitotal and total GRAPHS of a connected graph $G$.

Let $t_p(G)$ denote the number of paths in a graph $G$ and let $f:E\rightarrow \mathbb{Z}^+$ be an edge labeling of $G$. The weight of a path $P$ is the sum of the labels assigned to the edges of $P$. If the set of weights of the paths in $G$ is $\{1,2,3,\dots,t_p(G)\}$, then $f$ is called a Leech labeling of $G$ and a graph which admits a Leech labeling is called a Leech graph. In this paper, we prove that the complete bipartite GRAPHS $K_{2,n}$ and $K_{3,n}$ are not Leech GRAPHS and determine the maximum possible value that can be given to an edge in the Leech labeling of a cycle.

A coalition in a graph G = (V, E) consists of two disjoint sets V1 and V2 of vertices, such that neither V1 nor V2 is a dominating set, but the union V1 , V2 is a dominating set of G. A coalition partition in a graph G of order n = |V| is a vertex partition π,= {V1, V2, …, , Vk} such that every set Vi either is a dominating set consisting of a single vertex of degree n-1, or is not a dominating set but forms a coalition with another set Vj. Associated with every coalition partition π,of a graph G is a graph called the coalition graph of G with respect to π, , denoted CG(G,π, ), the vertices of which correspond one-to-one with the sets V1,V2,…, , Vk of π,and two vertices are adjacent in CG(G,π,) if and only if their corresponding sets in π,form a coalition. In this paper, we initiate the study of coalition GRAPHS and we show that every graph is a coalition graph.