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.

The zero forcing number of a graph is the minimum cardinality among all the zero forcing sets of a graph $G$.  The aim of this article is to compute the zero forcing number of complementary prism graphs.  Some bounds on the zero forcing number of complementary prism graphs are presented. The remainder of this article discusses the following result.  Let $G$ and $\overline{G }$ be connected graphs. Then $Z(G\overline{G})\leq n-1$ if and only if  there exists two vertices $v_i,v_j \in V(G)$ and $i\neq j$ such that, either $N(v_i) \subseteq N(v_j)$ or $N[v_i] \subseteq N[v_j]$ in $G$.

Let Γ = (V,E) be a simple and undirected graph. General power graph of Γ, shown by Pg(Γ), is a graph with the vertex set P(V (Γ))\ϕ. Also two distinct vertices of B and C are adjacent if and only if every b ∈ B is adjacent to every c ∈ C \{b} in Γ. In this paper, we consider general power graph related to graph Γ. Also we show that zero forcing number is equal to maximum nullity, for general power graph of some graphs.

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.

The zero forcing number of a graph $G$, denoted $Z(G)$, is a graph parameter  which is based on a color change rule that describes how to color the vertices. zero forcing is useful in several branches of science such as electrical engineering, computational complexity and quantum control.  In this paper, we investigate the zero forcing number for Cartesian products of some graphs. The main contribution of this paper is to introduce a new presentation of the Cartesian product of two complete bipartite graphs and to obtain the zero forcing number of these graphs.  We also introduce a purely graph theoretical method to prove $Z(K_n \Box K_m)=mn-m-n+2$.

Let 􀀀 be a graph whose each vertex is colored either white or black. If u is a black vertex of 􀀀 such that exactly one neighbor v of u is white, then u changes the color of v to black. A zero forcing set for a graph 􀀀 is a subset of vertices Z  V (􀀀 ) such that if initially the vertices in Z are colored black and the remaining vertices are colored white, then Z changes the color of all vertices in 􀀀 to black. The zero forcing number of 􀀀 is the minimum of jZj over all zero forcing sets for 􀀀 and is denoted by Z(􀀀 )...

The zero forcing number Z(G) of a graph G is the minimum cardinality of a set S with colored (black) vertices which forces the set V (G) to be colored (black) after some times. \color change rule": a white vertex is changed to a black vertex when it is the only white neighbor of a black vertex. In this case, we say that the black vertex forces the white vertex. We investigate here the concept of connected zero forcing set and connected zero forcing number. We discusses this subject for special graphs and some products of graphs. Also we introduce the connected propagation time. Graphs with extreme minimum connected propagation times and maximum propagation times |G| -1 and |G| -2 are characterized.

In this article we study the zero forcing number of Generalized Sierpi nski graphs S(G; t). More precisely, we obtain a general lower bound on the zero forcing number of S(G; t) and we show that this bound is tight. In particular, we consider the cases in which the base graph G is a star, path, a cycle or a complete graph.