Signed zero forcing number and controllability for a networks system with a directed hypercube

The controllability for complex network system is to find the minimum number of leaders for the network system to achieve effective control of the global networks. In this paper, the problem of controllability of the directed network for a family of matrices carrying the structure under directed hypercube is considered. The relationship between the minimum number of leaders for the directed network system and the number of the signed zero forcing set is established. The minimum number of leaders of the directed networks system under a directed hypercube is obtained by computing the zero forcing number of a signed graph.

All Graphs with a Failed Zero Forcing Number of Two

Given a graph G, the zero forcing number of G, Z(G), is the smallest cardinality of any set S of vertices on which repeated applications of the forcing rule results in all vertices being in S. The forcing rule is: if a vertex v is in S, and exactly one neighbor u of v is not in S, then u is added to S in the next iteration. Zero forcing numbers have attracted great interest over the past 15 years and have been well studied. In this paper, we investigate the largest size of a set S that does not force all of the vertices in a graph to be in S. This quantity is known as the failed zero forcing number of a graph and will be denoted by F(G). We present new results involving this parameter. In particular, we completely characterize all graphs G where F(G)=2, solving a problem posed in 2015 by Fetcie, Jacob, and Saavedra.

Techniques for determining equality of the maximum nullity and the zero forcing number of a graph

It is known that the zero forcing number of a graph is an upper bound for the maximum nullity of the graph (see [AIM Minimum Rank - Special Graphs Work Group (F. Barioli, W. Barrett, S. Butler, S. Cioab$\breve{\text{a}}$, D. Cvetkovi$\acute{\text{c}}$, S. Fallat, C. Godsil, W. Haemers, L. Hogben, R. Mikkelson, S. Narayan, O. Pryporova, I. Sciriha, W. So, D. Stevanovi$\acute{\text{c}}$, H. van der Holst, K. Vander Meulen, and A. Wangsness). Linear Algebra Appl., 428(7):1628--1648, 2008]). In this paper, we search for characteristics of a graph that guarantee the maximum nullity of the graph and the zero forcing number of the graph are the same by studying a variety of graph parameters that give lower bounds on the maximum nullity of a graph. Inparticular, we introduce a new graph parameter which acts as a lower bound for the maximum nullity of the graph. As a result, we show that the Aztec Diamond graph's maximum nullity and zero forcing number are the same. Other graph parameters that are considered are a Colin de Verdiére type parameter and vertex connectivity. We also use matrices, such as a divisor matrix of a graph and an equitable partition of the adjacency matrix of a graph, to establish a lower bound for the nullity of the graph's adjacency matrix.

Zero Forcing Number of Some Families of Graphs

The work is devoted to the study of the zero forcing number of some families of graphs. The concept of zero forcing is a relatively new research topic in discrete mathematics, which already has some practical applications, in particular, is used in studies of the minimum rank of the matrices of adjacent graphs. The zero forcing process is an example of the spreading process on graphs. Such processes are interesting not only in terms of mathematical and computer research, but also interesting and are used to model technical or social processes in other areas: statistical mechanics, physics, analysis of social networks, and so on. Let the vertices of the graph G be considered white, except for a certain set of S black vertices. We will repaint the vertices of the graph from white to black, using a certain rule.Colour change rule: A white vertex turns black if it is the only white vertex adjacent to the black vertex.[5] The zero forcing number Z(G) of the graph G is the minimum cardinality of the set of black vertices S required to convert all vertices of the graph G to black in a finite number of steps using the ”colour change rule”.It is known [10] that for any graph G, its zero forcing number cannot be less than the minimum degree of its vertices. Such and other already known facts became the basis for finding the zero forcing number for two given below families of graphs:A gear graph, denoted W2,n is a graph obtained by inserting an extra vertex between each pair of adjacent vertices on the perimeter of a wheel graph Wn. Thus, W2,n has 2n + 1 vertices and 3n edges.A prism graph, denoted Yn, or in general case Ym,n, and sometimes also called a circular ladder graph, is a graph corresponding to the skeleton of an n-prism.A wheel graph, denoted Wn is a graph formed by connecting a single universal vertex to all vertices of a cycle of length n.In this article some known results are reviewed, there is also a definition, proof and some examples of the zero forcing number and the zero forcing process of gear graphs and prism graphs.

Maximum nullity and zero forcing of circulant graphs

The zero forcing number of a graph has been applied to communication complexity, electrical power grid monitoring, and some inverse eigenvalue problems. It is well-known that the zero forcing number of a graph provides a lower bound on the minimum rank of a graph. In this paper we bound and characterize the zero forcing number of various circulant graphs, including families of bipartite circulants, as well as all cubic circulants. We extend the definition of the Möbius ladder to a type of torus product to obtain bounds on the minimum rank and the maximum nullity on these products. We obtain equality for torus products by employing orthogonal Hankel matrices. In fact, in every circulant graph for which we have determined these numbers, the maximum nullity equals the zero forcing number. It is an open question whether this holds for all circulant graphs.

Zero forcing number of fuzzy graphs with application

We introduce and study forcing number for fuzzy graphs. Also, we compute zero forcing numbers for some classes of graphs and extend this concept to fuzzy graphs. In this regard we obtain upper bounds for zero forcing of some classes of fuzzy graphs. We will proceed to obtain a new algorithm to computing zero forcing set and finding a formula for zero forcing number, and by some examples we illustrate these notions. Finally, we introduce some applications of fuzzy zero forcing in medical treatments.