The current study aims to establish a connection between graphs and automata theory, which apparently demonstrate different mathematical structures. Through searching out some properties of one of these structures, we try to find some new properties of the other structure as well. This will result in obtaining some unknown properties. At first, a novel automaton called zero-forcing (Z-F) finite automata is defined 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-finite automata will be obtained. In addition, the language and the closure properties of Z-F-finite 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-finite automata are presented. Further, it is shown that there is not any finite graph such that f be a part of the language of its Z-F-finite automata. Actually, it is proved that for every given graph, the Z-F-finite 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-finite automata of it has a weak closed trail containing all edges. Some examples are also given to clarify these new notions. تفاصيل المقالة

This paper introduces the notion of the combined product of two RL-graphs while it is an RL-graph. It is stated that the combined product has commutative properties, i.e., G ⊠ H and H ⊠ G are two isomorphic RL-graphs. Moreover, it is shown under a theorem that two isomorphic RL-graphs G and G′ and two isomorphic RL-graphs H and H′ have isomorphic combined products G ⊠ H and G′ ⊠ H′. Further, it is investigated the relationships between these graphs and their operations by some notions such as strong RL-graph, regular RL- graph, and totally regular RL-graph. Afterward, it is displayed in a theorem that the combined product of two regular (α-regular, complete, connected) RL-graphs is a regular (α-regular, complete, connected) RL-graph. It is also shown in theorems what properties certain types of RL-graphs combined will have. Also, these notions and theorems are clarified by some examples. The combined product of two RL-graphs has many applications in various fields, such as probability sciences, urban planning, etc. In this article, only two of these applications, which determine the impact of effective factors on people’s quality of life and factors effective in raising the production of a factory, are stated and they are clarified by an example تفاصيل المقالة