The current study aimed to describe the notion of the L-Graph, which is constructed on a residuated lattice, presenting the idea of the maximal product of two L-Graphs.    Moreover, the applied algorithm demonstrated its efficiency to a great extent to improve the educational system.    The results showed that the maximal product of both G and H and reversely, H    and  G are isomorphic L-Graphs.    The maximal product of two L-Graph automata was represented as a new notion,    followed by investigating their self-sufficiency conditions in two different modes.    In addition,  A(Z(G))\star~A(Z(H)) and A(Z(H))\star~A(Z(G))  were proved as two isomorphic-related L-Graph automata considering the residuated lattice properties.    Then, correlations among A(Z(G))\star A(Z(H)),  A(Z(G)),  and A(Z(H)) behaviors were explained.    As a result, some theorems of the relationship between L-Graph automata and their maximal product were introduced.    Subsequently, some related theorems were proved, and several examples were provided to illustrate these new notions. Further, an application of the maximal product of two related L-Graph automata was expressed in the factors affecting the spared of the coronavirus,  and accordingly,    some solutions were suggested to reduce the spread of the virus.

In this note, we show that automata theory is a suitable tool for analyzing monopoly-forcing processes. Also, we present the notion of mono-forcing automata by using the monopoly-forcing set for Graphs. Moreover, we prove that mono-forcing automata accept more languages than zero-forcing finite automata also, we show that all results in zero-forcing finite automata for complete Graphs are established for mono-forcing automata. We examine and deliberate on the language associated with mono-forcing automata for certain specified Graphs. Also, we present the style of words that can be recognized with mono-forcing automata. Additionally, we delineate the types of words identifiable by mono-forcing automata. We also describe the configuration of Graphs from which mono-forcing automata emerge, generating specific languages. Several examples are provided to elucidate these concepts.

In this paper, a new criterion is introduced for solving optimization problems on stochastic Graphs- as a model of computer networks-by stochastic learning automata. This proposed method, because of considering estimated variance of response of environment, can better adaptation to changes of environment. As a result, the proposed method can produce better response to learning automata actions.The proposed method, by entering a noise, can avoid learning automata being stuck at a local optimum point. Our simulation shows that this proposed method can be improve the convergence rate of automata-based algorithm.

All the social networks can be modeled as a Graph, where each roles as vertex and each relation roles as an edge. The Graph can be show as G = [V; E], where V is the set of vertices and E is the set of edges. All social networks can be segmented to K groups, where there are members in each group with same features. In each group each person knows other individuals and is in touch with them. In this study, the main goal is introducing a new approach for detecting these groups and minimizing the number of these groups using a cellular automat algorithm. There are two types of social networks, containing simulated social network and real social network. The results show that the introduced method has a great potential to signi cantly reduce the number of colors assigned and running time of the program.

The current study aims to establish a connection between Graphs and automata theory, which apparently demonstrate di erent mathematical structures. Through searching out some properties of one of these structures, we try to  nd some new properties of the other structure as well. This will result in obtaining some unknown properties. At  rst, a novel automaton called zero-forcing (Z-F)  nite automata is de ned according to the notion of a zero-forcing set of a Graph. It is shown that for a given Graph and for some zero forcing sets, various Z-F- nite automata will be obtained. In addition, the language and the closure properties of Z-F- nite automata, in particular; union, connection, and serial connection are studied. Moreover, considering some properties of Graphs such as the closed trail, connected and complete; some new features for Z-F- nite automata are presented. Further, it is shown that there is not any  nite Graph such that f be a part of the language of its Z-F- nite automata. Actually, it is proved that for every given Graph, the Z-F- nite automata of it does not show any closed trail containing all edges for every zero forcing set, but if the Graph G has been a closed trail containing all edges, then the Z-F- nite automata of it has a weak closed trail containing all edges. Some examples are also given to clarify these new notions.

In this paper a new structure of learning automata which is called as extended distributed learning automata (eDLA) is introduced. A new eDLA-based iterative sampling method for finding optimal sub-Graph in stochastic Graphs is proposed. Some mathematical analysis of the proposed algorithm is presented and the convergence property of the algorithm is studied. Our study shows that the proposed algorithm can be converge to the optimal sub-Graph.

In this paper, the concepts of somewhat fuzzy automata continuous functions and somewhat fuzzy automata open functions in fuzzy automata topological spaces are introduced and some interesting properties of these functions are studied. In this connection, the concepts of fuzzy automata resolvable spaces and fuzzy automata irresolvable spaces are also introduced and their properties are studied.