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Families of Graphs with Maximum Nullity Equal to Zero Forcing Number Mathematics Publications Mathematics 2-2-2018 Families of graphs with maximum nullity equal to zero forcing number Joseph S. Alameda Iowa State University, [email protected] Emelie Curl Iowa State University, [email protected] Armando Grez Iowa State University, [email protected] Leslie Hogben Iowa State University, [email protected] O'Neill Kingston Iowa State University, [email protected] See next page for additional authors Follow this and additional works at: https://lib.dr.iastate.edu/math_pubs Part of the Algebra Commons, and the Discrete Mathematics and Combinatorics Commons The complete bibliographic information for this item can be found at https://lib.dr.iastate.edu/ math_pubs/172. For information on how to cite this item, please visit http://lib.dr.iastate.edu/ howtocite.html. This Article is brought to you for free and open access by the Mathematics at Iowa State University Digital Repository. It has been accepted for inclusion in Mathematics Publications by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Families of graphs with maximum nullity equal to zero forcing number Abstract The maximum nullity of a simple graph G, denoted M(G), is the largest possible nullity over all symmetric real matrices whose ijth entry is nonzero exactly when fi, jg is an edge in G for i =6 j, and the iith entry is any real number. The zero forcing number of a simple graph G, denoted Z(G), is the minimum number of blue vertices needed to force all vertices of the graph blue by applying the color change rule. This research is motivated by the longstanding question of characterizing graphs G for which M(G) = Z(G). The following conjecture was proposed at the 2017 AIM workshop Zero forcing and its applications: If G is a bipartite 3- semiregular graph, then M(G) = Z(G). A counterexample was found by J. C.-H. Lin but questions remained as to which bipartite 3-semiregular graphs have M(G) = Z(G). We use various tools to find bipartite families of graphs with regularity properties for which the maximum nullity is equal to the zero forcing number; most are bipartite 3-semiregular. In particular, we use the techniques of twinning and vertex sums to form new families of graphs for which M(G) = Z(G) and we additionally establish M(G) = Z(G) for certain Generalized Petersen graphs. Keywords zero forcing number, maximum nullity, semiregular bipartite graph, Generalized Petersen graph Disciplines Algebra | Discrete Mathematics and Combinatorics | Mathematics Comments This article is published as Alameda, Joseph S., Emelie Curl, Armando Grez, Leslie Hogben, Alex Schulte, Derek Young, and Michael Young. "Families of graphs with maximum nullity equal to zero forcing number." Special Matrices 6, no. 1 (2018): 56-67. DOI: 10.1515/spma-2018-0006. Posted with permission. Creative Commons License This work is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International License. Authors Joseph S. Alameda, Emelie Curl, Armando Grez, Leslie Hogben, O'Neill Kingston, Alex Schulte, Derek Young, and Michael Young This article is available at Iowa State University Digital Repository: https://lib.dr.iastate.edu/math_pubs/172 Spec. Matrices 2018; 6:56–67 Research Article Open Access Special Issue on Linear Algebra and its Applications (ICLAA2017) Joseph S. Alameda, Emelie Curl, Armando Grez, Leslie Hogben, O’Neill Kingston, Alex Schulte, Derek Young*, and Michael Young Families of graphs with maximum nullity equal to zero forcing number https://doi.org/10.1515/spma-2018-0006 Received September 27, 2017; accepted January 19, 2018 Abstract: The maximum nullity of a simple graph G, denoted M(G), is the largest possible nullity over all symmetric real matrices whose ijth entry is nonzero exactly when fi, jg is an edge in G for i ≠ j, and the iith entry is any real number. The zero forcing number of a simple graph G, denoted Z(G), is the minimum number of blue vertices needed to force all vertices of the graph blue by applying the color change rule. This research is motivated by the longstanding question of characterizing graphs G for which M(G) = Z(G). The following conjecture was proposed at the 2017 AIM workshop Zero forcing and its applications: If G is a bipartite 3- semiregular graph, then M(G) = Z(G). A counterexample was found by J. C.-H. Lin but questions remained as to which bipartite 3-semiregular graphs have M(G) = Z(G). We use various tools to nd bipartite families of graphs with regularity properties for which the maximum nullity is equal to the zero forcing number; most are bipartite 3-semiregular. In particular, we use the techniques of twinning and vertex sums to form new families of graphs for which M(G) = Z(G) and we additionally establish M(G) = Z(G) for certain Generalized Petersen graphs. Keywords: zero forcing number; maximum nullity; semiregular bipartite graph; Generalized Petersen graph MSC: 05C50, 15A03, 15A18 1 Introduction Let V be a nite nonempty set. A graph G = (V, E) is a pair of sets such that E is a set of two element subsets of V. The elements of V are called vertices and the elements of E are called edges. The vertex set of a graph G is often denoted by V(G) and the edge set by E(G). The order of G is the cardinality of V(G) and the size of G is the cardinality of E(G). An edge fu, vg is usually written as uv. Vertices u and v are adjacent in G if uv 2 E(G). A vertex u is a neighbor of v if uv 2 E(G). The neighborhood of v, denoted by N(v), is the set of neighbors of v. The adjacency matrix of G is A(G) = [aij] where aij = 1 if ij 2 E(G) and aij = 0 otherwise. For a graph G, the set of symmetric matrices of G over R, denoted by S(G), is the set of real symmetric matrices A = [aij] such that aij is non-zero if ij 2 E(G), aij is any real number if i = j, and aij is 0 otherwise. The minimum rank of G is Joseph S. Alameda: Department of Mathematics, Iowa State University, Ames, IA 50011, USA, E-mail: [email protected] Emelie Curl: Department of Mathematics, Iowa State University, Ames, IA 50011, USA, E-mail: [email protected] Armando Grez: Department of Mathematics, Iowa State University, Ames, IA 50011, USA, E-mail: [email protected] Leslie Hogben: Department of Mathematics, Iowa State University, Ames, IA 50011, USA, E-mail: [email protected], and American Institute of Mathematics, 600 E. Brokaw Road, San Jose, CA 95112, USA, E-mail: [email protected] O’Neill Kingston: Department of Mathematics, Iowa State University, Ames, IA 50011, USA, E-mail: [email protected] Alex Schulte: Department of Mathematics, Iowa State University, Ames, IA 50011, USA, E-mail: [email protected] *Corresponding Author: Derek Young: Department of Mathematics, Iowa State University, Ames, IA 50011, USA, E-mail: [email protected] Michael Young: Department of Mathematics, Iowa State University, Ames, IA 50011, USA, E-mail: [email protected] Open Access. © 2018 Joseph S. Alameda et al., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 License. Brought to you by | Iowa State University Authenticated Download Date | 4/9/18 7:06 PM Families of graphs with maximum nullity equal to zero forcing number Ë 57 mr(G) = minfrank(A) | A 2 S(G)g. The maximum nullity of G is dened as M(G) = maxfnull(A) | A 2 S(G)g. Observe that mr(G) + M(G) = jV(G)j where j · j denotes cardinality. In order to introduce zero forcing we will dene the color change rule as follows: Suppose a graph G has every vertex colored either blue or white, and b is a blue vertex. If b has exactly one white neighbor, w, then we change the color of w to blue. We say that b forces w, and this can be denoted by b ! w. Let S ⊆ V(G). The nal coloring of S is the result of initially coloring every vertex in S blue and every vertex in V(G)\S white, and then applying the color-change rule until no more color changes can be made. Note that the order in which forces occur does not aect the nal coloring of G. The set S is called a zero forcing set if the nal coloring of S is all blue. The zero forcing number of a graph G is Z(G) = minfjSj | S is a zero forcing set of Gg. It is well known from [1] that M(G) ≤ Z(G). This paper addresses the longstanding question of determining graphs G for which M(G) = Z(G) (see [1, Question 1]). The degree of v, deg(v), is the number of edges incident to v. Note that deg(v) = jN(v)j. A vertex with degree equal to 1 is called a leaf. The minimum degree of G is δ(G) = minfdeg(v) | v 2 V(G)g and the maximum degree of G is ∆(G) = maxfdeg(v) | v 2 V(G)g. A graph G is regular if ∆(G) = δ(G). We call G cubic if ∆(G) = 3 = δ(G). A graph G is bipartite if V(G) can be partitioned into two sets X and Y such that N(x) ⊆ Y and N(y) ⊆ X for x 2 X and y 2 Y; the partition of vertices can be denoted by G(X, Y).
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