In this article the square of directed graph will be discussed. H2 is the square of graph H which with adding edges joins two separated vertexes with the distance of 2 (when H is a un directed graph). In this article the second power of directed graphs is studied for the first time and related theorem to this subject and some examples to explain it will be mentioned.

Prime graph of a ring R is a graph whose vertex set is the whole set R any any two elements x and y of R are adjacent in the graph if and only if xRy = 0 or yRx = 0. Prime graph of a ring is denoted by PG(R). Directed prime graphs for non-commutative rings and connectivity in the graph are studied in the present paper. The diameter and girth of this graph are also studied in the paper.

Prediction of the future behavior of the stock market has always attracted researchers' attention as an important challenge in the field of machine learning. In recent years deep learning methods have been successfully applied in this domain to improve prediction performance. Previous studies have demonstrated that aggregating information from related stocks can improve the performance of prediction. However, the capacity of modeling the stocks relations as directed graphs and the power of sophisticated graph embedding techniques such as Graph Attention Networks have not been exploited so far for prediction in this domain. In this work, we introduce a framework called DeepNet that creates a directed graph representing how useful the data from each stock can be for improving the prediction accuracy of any other stocks. DeepNet then applies Graph Attention Network to extract a useful representation for each node by aggregating information from its neighbors, while the optimal amount of each neighbor's contribution is learned during the training phase. We have developed a novel Graph Attention Network model called DGAT that is able to define unequal contribution values for each pair of adjacent nodes in a directed graph. Our evaluation experiments on the Tehran Stock Exchange data show that the introduced prediction model outperforms the state-of-the-art baseline algorithms in terms of accuracy and MCC measures.

Deza digraphs were introduced in 2003 by Zhang and Wang as directed graph version of Deza graphs, that also generalize the notion of directed strongly regular graphs. In this paper, we give several new constructions of Deza digraphs. Further, we introduce twin and Siamese twin (directed) Deza graphs and construct several examples. Finally, we study a variation of directed Deza graphs and provide a construction from finite fields.

In this paper, the problem of self-tuning of coupling parameters in multi-agent systems is considered. Agent dynamics are described by a discrete-time double integrator with time-varying nonhomogeneous input gain. The coupling parameters defining the strength of agents interactions are locally self-tuning by each node based on the velocities of its neighbors. The cost function is equal to the square of the local error between the agent velocity and the weighted average of the velocities of interacting neighbors. So, the proposed algorithm is the normalized gradient algorithm which is minimized the square of the local error between the agent velocity and the one step delayed average of the velocities of its neighbors. Provided that the underlying graph is strongly connected, it is shown that the sequence of the inter-agent coupling parameters generated by the proposed algorithm is convergent. Also, assuming the suitable initial condition on coupling parameters, it is proved that the network achieves average consensus. In other words, the agent velocities converge toward the average of the initial velocities values. Furthermore, the distance among agents converges to a finite limit. Simulation results illustrate effectiveness of the proposed method.

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.

This paper, studies the controller design for containment problem for a class of multiagent systems with identical time-invariant continuous-time nonlinear dynamics and fixed directed communication graph. In this problem, the Lyapunov stability theorem, the graph theory and matrix linear inequality are used. The agents are divided into two groups of leaders and followers. In containment problem, all the followers are controlled under which will asymptotically converge to the convex hull spanned by the leaders so distributed communication protocol with fixed time delay is considered and four-step algorithm is proposed for obtaining parameters and gain matrix. The above case is proved to be sufficient condition under theorem. To illustrate the reliability and efficiency of the proposed method, numerical example with simulations are presented.

To date, the feasibility of constructing a complete graph invariant in polynomial time remains uncertain. Therefore, developing fast algorithms for checking non-isomorphism, including heuristic ones, is crucial. Successful implementation of these heuristics involves modifying existing graph invariants and creating new ones, both of which are still pertinent. Many existing invariants enable the distinction of a large number of graphs in real time.This paper introduces an invariant specifically for tournaments, a type of directed graph. Tournaments are interesting because the number of different tournaments, given a fixed order of vertices, matches the number of undirected graphs with the same fixed order. The proposed invariant considers all possible tournaments formed by subsets of vertices from the given digraph with the same set of arcs. For each subset tournament, standard places are calculated and summed to determine the final vertex points, which constitute the new invariant.Our calculations reveal that the new invariant differs from the most natural tournament invariant, which assigns points to each participant. Initial computational experiments show that the smallest pair correlation between sequences representing these two invariants is observed at dimension 15.

Matrix completion problem has gathered a lot of attention in recent years. In the matrix completion problem, the goal is to recover a low-rank matrix from a subset of its entries. The graph matrix completion was introduced based on the fact that the relation between rows (or columns) of a matrix can be modeled as a graph structure. The graph matrix completion problem is formulated by adding the graph total variation term to the objective function of matrix completion problem. However; in practice, the observed data is noisy and contains outliers. Outlier data is defined as part of the observed that are different than other parts and are not consistent with the data structure. In this paper, we apply graph total variation based on the directed Laplacian and propose a new method for graph matrix completion. We introduce a new method called GMCO-DL for the case where both noise and outliers exist in observations. Simulation results show outstanding results for the proposed method in terms of estimation error.