Available online at http://ijim.srbiau.ac.ir/ Int. J. Industrial Mathematics (ISSN 2008-5621) Vol. 13, No. 4, 2021 Article ID IJIM-1493, 12 pages Research Article Zero-forcing Finite Automata M. Shamsizadeh ∗† , M. M. Zahedi ‡ , M. Golmohamadian § , KH. Abolpour ¶ Received Date: 2021-02-13 Revised Date: 2021-06-21 Accepted Date: 2021-08-01 ————————————————————————————————– Abstract The current study aims to establish a connection between graphs and automata theory, which appar- ently 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. Keywords : Graph; Zero forcing set; Automata; Graph automata; Language of automata. —————————————————————————————————– 1 Introduction A n automaton is a mathematical theory in in- vestigates behavior, structure and their re- lationship to discrete systems. Directable au- tomata were introduced by P. H. Strake in [24] ∗ Corresponding author. shamsizadeh.m@gmail.com, Tel:+98(917)3084435. † Department of Mathematics, Behbahan Khatam Alan- bia University of Technology, Khouzestan, Iran. ‡ Department of Mathematics, Graduate University of Advanced Technology, Kerman, Iran. § epartment of Mathematics, Tarbiat Modares Univer- sity, Tehran, Iran. ¶ Department of Mathematics, Shiraz Branch, Islamic Azad University, Shiraz, Iran. and J. Cerny in [10], and also definite automata by S. C. Kleene in [16]. Today, finite automata have many applications in plenty of areas of com- puter science such as databases, functional lan- guages, bisimulation, and biology, for more infor- mation see [1, 9, 11, 14, 18, 21, 22, 23]. For more than one hundred years, the develop- ment of graph theory was inspired and guided mainly by the Four-Colour Conjecture. The resolution of the conjecture by K. Appel and W. Haken in 1976, the year in which our first book Graph Theory with Applications appeared, marked a turning point in its history. Since then, the subject has experienced explosive growth, due in large measure to its role as an essential struc- ture underpinning modern applied mathematics. 477