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.

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.

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.