Families of graphs with maximum nullity equal to zero forcing number

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 3semiregular 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.


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 {u, v} is usually written as uv. Vertices u and v are adjacent in G if uv ∈ E(G). A vertex u is a neighbor of v if uv ∈ 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) = [a ij ] where a ij = if ij ∈ E(G) and a ij = 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 = [a ij ] such that a ij is non-zero if ij ∈ E(G), a ij is any real number if i = j, and a ij is otherwise. The minimum rank of G is mr(G) = min{rank(A) | A ∈ S(G)}. The maximum nullity of G is de ned as M(G) = max{null(A) | A ∈ S(G)}. Observe that mr(G) + M(G) = |V(G)| where | · | denotes cardinality.
In order to introduce zero forcing we will de ne 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 a ect 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) = min{|S| | S is a zero forcing set of G}. 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) = |N(v)|. A vertex with degree equal to 1 is called a leaf.
can be partitioned into two sets X and Y such that N(x) ⊆ Y and N(y) ⊆ X for x ∈ X and y ∈ Y; the partition of vertices can be denoted by G (X, Y).
. That is, an induced graph is one obtained by deleting vertices and incident edges.
A graph G is a path on n vertices, denoted by A complete graph on n vertices, denoted by Kn, is a graph of order n with all possible edges between its vertices. A complete bipartite graph is a bipartite graph with all possible edges between the two parts and is denoted by Kn,m where n and m are the orders of the two parts.
The union of graphs G = (V , E ) and can be expressed as a vertex sum at v.
At the 2017 American Institute of Mathematics workshop Zero forcing and its applications, it was conjectured that M(G) = Z(G) if G is a bipartite -semiregular graph [2]. It is known that all bipartite 1-semiregular and bipartite 2-semiregular graphs satisfy M(G) = Z(G), but not all bipartite graphs have M(G) = Z(G). A counterexample to the conjecture was found by J.C.-H. Lin [8] (see Example 1.1), but questions remain as to which bipartite 3-semiregular graphs have M(G) = Z(G). In this paper, we construct bipartite families of graphs with regularity properties for which the maximum nullity is equal to the zero forcing number. In Section 2 we establish that M(G) = Z(G) for many Generalized Petersen graphs (these graphs are all cubic and some are bipartite). In Section 3 we develop expansion techniques that preserve M(G) = Z(G) and apply them to families of graphs in Section 4. In particular, we use twinning and vertex sums to form new families of bipartite graphs for which M(G) = Z(G); some of these are 3-semiregular. In Section 5 we show that M(G) = Z(G) for two well-known cubic graphs.  [10] showed that M(G) ≤ P(G) for an outerplanar graph G. Since L is outerplanar and P(L) = , M(L) ≤ . By use of software [6] it is straightforward to verify that Z(L) = .

Generalized Petersen Graphs
In this section, we present results on the zero forcing number and maximum nullity of the Generalized Petersen (abbreviated GP) graphs, including exact values for a few subfamilies of GP graphs. We rst de ne the aforementioned family using notation similar to that in [11]. For n ≥ and < k ≤ n− , the Generalized Petersen (GP) graph P(n, k) is the graph with vertex set The graph P(n, k) is a cubic graph consisting of an outside cycle on vertices u , . . . u n− with a perfect matching connecting this cycle to one or more inner cycles on vertices v , . . . , v n− where the vertices on the inner and outer cycles are ordered counterclockwise (see Figure 2). In the literature, the term Generalized Petersen graph often requires that n and k be relatively prime. However, for the purposes of this paper we do not require this condition. Note that in the above de nition k is restricted as < k ≤ n− . This is because the symmetry of the GP graphs gives us that for n− < < n, P(n, ) is isomorphic to P(n, n − ), and < n − < n− . Some GP graphs are known by di erent names such as the n-Prism, which is P(n, ), and the well-known Petersen graph P( , ). It is known that M(P(n, )) = Z(P(n, )) = for n ≥ , M(P( , )) = Z(P( , )) = , and M(P( , )) = Z(P( , )) = [1].

Remark 2.1.
A GP graph is bipartite if and only if n is even and k is odd. This is because P(n, k) contains an odd cycle if n is odd or if k is even, and a graph is bipartite if and only if it does not contain an odd cycle.
Initially, the vertices in S can force the vertices in S because u i forces v i for i = , . . . , k− . Then u n−i+ forces u n−i+ and v k−i+ forces v n−i+ , for ≤ i ≤ n + . Since all vertices are eventually forced, Z(P(n, k)) ≤ |S | = k + .
Brought to you by | Iowa State University Authenticated Download Date | 4/9/18 7:06 PM Since some but not all of our arithmetic is modular, we de ne notation for the residue mod n of an integer by In this section we index the rows and columns of matrices and vectors starting with zero. To form the adjacency matrix of a GP graph, we order the vertices as u , u , u , . . . , u i , . . . , u n− followed by the vertices v , v , v , . . . , v i , . . . , v n− . In the GP graph P(n, k), the neighbors of an outer cycle vertex Thus, the adjacency matrix is a block matrix of the following form where A(Cn) is the adjacency matrix of the cycle on n vertices and the matrix A (Cn) is a matrix with 1's on the kth and (n − k)th super and subdiagonals and zeros elsewhere (i.e., A (Cn) is the adjacency matrix of the inner cycle(s)).
Next we establish a technical lemma about eigenvalue multiplicities of GP graphs and then use it to show certain subfamilies of GP graphs have maximum nullity equal to zero forcing number. Lemma 2.3. Let n ≥ , < k ≤ n− , n = nr for r ≥ , and k satis es k n = k and < k ≤ n − . Suppose λ is an eigenvalue of A(P(n, k)) with multiplicity m. Then, λ is an eigenvalue of A(P(n , k )) with multiplicity at least m.
Proof. For each eigenvector of A(P(n, k)) for λ, we construct an eigenvector of A(P(n , k )) for λ. Since by construction independent eigenvectors of A(P(n, k)) yield independent eigenvectors of A(P(n , k )), this shows λ is an eigenvalue of multiplicity at least m for A(P(n , k )). Suppose x = [x j ] is an eigenvector of A(P(n, k)) for eigenvalue λ. By considering row j for ≤ j ≤ n − and using that j = (j) n we have Brought to you by | Iowa State University Authenticated Download Date | 4/9/18 7:06 PM Considering row j for n ≤ j ≤ n − and using that j − n = (j) n and (j − n) n = (j) n yields (2) We de nex Observe that independence of a set of n-vectors x, y, . . . guarantees the independence of the set of n -vectorŝ x,ŷ, . . . just constructed. We show thatx is an eigenvector of A(P(n , k )) for eigenvalue λ. Note thatx j = x (j) n for j = , . . . , n − , andx j = x n+(j−n ) n = x n+(j) n for j = n , . . . , n − . Observe that (c) n n = (c) n for all integers c. Thus for j = , . . . , n − , Thus, we conclude A(P(n , k ))x = λx.
Theorem 2.4. Let n ≥ and < k ≤ n− . Suppose there is an eigenvalue λ of A(P(n, k)) with multiplicity m. Then λ is an eigenvalue of A(P(nr, k)) with multiplicity at least m. If m = k + , then M(P(nr, k)) = Z(P(nr, k)) = k + for all integers r ≥ . In particular, the following Generalized Petersen graphs have maximum nullity equal to zero forcing number for r ≥ : (a) M(P( r, )) = Z(P( r, )) = .

Expansion Procedures
In this section, we introduce expansion procedures that determine the maximum nullity and minimum rank of graphs with special characteristics. In a graph G, vertices v and w that have the same set of neighbors Proof. Since null(A) = M(G), we have rank(A) = mr(G). By Proposition 3.1 there is a matrix A ∈ S(twin(G, v, k)) with rank(A ) = mr(twin(G, v, k)) = mr(G) and a vv = . Since twin(G, v, k)) has k more vertices than G, M(twin(G, v, k)) = M(G) + k. Since we can simply color an independent twin blue and add it to a given zero forcing set, we know that Z(twin (G, v, k)) ≤ Z(G) + k. Thus, M(G) + k = M(twin(G, v, k)) ≤ Z(twin (G, v, k)) ≤ Z(G) + k = M(G) + k.

Families of bipartite graphs constructed by expansion
In this section we apply results from the previous section to construct families of bipartite graphs, some of which are 3-semiregular.

. Vertex sums of bat graphs
De ne the family of bat graphs as follows: The basic bat graph, which is denoted by B , is the graph given by a C with a leaf appended to each of two non-adjacent vertices. See Figure 3. This graph has M(B ) = Z(B ) = with its adjacency matrix being of minimum rank. The set consisting of one leaf and one degree two vertex is a minimum zero forcing set. In the forcing process, the other leaf does not force. Call the degree 3 vertices x and x , the degree 2 vertices y and y , and the leaves y and y . Any graph constructed by a sequence of twinning operations applied to x or x is a bat graph.  Proof. For the adjacency matrix of B , rank(A(B )) = = mr(B ) and all diagonal elements are zero, so we may apply the twinning operation described in Corollary 3.2.
Note that any bat graph is a 3-semiregular bipartite graph. Once at least two independent twins have been added to the X set of V(B ), the resulting graph has |X| ≥ |Y|. In particular, y or y may be in a minimum zero forcing set of any bat graph, and a reversal of that set will include whichever of the two was not in the original set. Similarly, if G is a vertex sum of bat graphs as described above, additional bat graphs may be appended to the y or y vertices of the "bat subgraphs" not already involved in a vertex sum while preserving the property that maximum nullity equals zero forcing number, allowing the construction of chains of bats, called bat chains, of arbitrary length. See Figure 4.

. Jewel necklace graphs
In this section, we introduce a family of graphs called jewel necklaces and show that maximum nullity equals the zero forcing number for these graphs. De ne an r-jewel to be a Kr,r where one edge is deleted. De ne an s, r-jewel necklace, Js,r, for s ≥ and r ≥ to be a cycle of jewel graphs connected appropriately by edges between vertices v that have deg(v) = r − . We require that r ≥ and s ≥ because otherwise we would be examining a cycle (r = ), s copies of P (r = ), or a Kr,r (s = ), for which maximum nullity and zero forcing number are known. In this process of connecting s copies of r-jewels, exactly one connecting edge is incident with each vertex of degree r − , so Js,r is r-regular. Figure 5 shows the general form of Js,r. Note that the order of Js,r is sr and this graph is bipartite.    Figure 6. We see that vertices ( , ) and ( , r) force vertices ( , r − ) and ( , r − ), respectively. Once these vertices are forced, vertices ( , ) and ( , r − ) force vertices ( , ) and ( , r − ), respectively. Now ( , ) and ( , r) can force the nal vertices. This process can be generalized for Js,r, with the zero forcing set shown Figure 5, with one jewel having its end vertices colored and the others not.

Other cubic graphs with M(G) = Z(G)
In this section we show that M(G) = Z(G) for two other well-known cubic graphs. The Bidiakis cube, BC, is a -vertex graph consisting of a cube in which two opposite faces (which we call top and bottom) have edges drawn across them which connect the centers of opposite sides of the faces in such a way that the orientation of the edges added on top and bottom are perpendicular to each other. Note that the Bidiakis cube is also isomorphic to a C with edges added between three pairs of vertices on opposite sides of the cycle. This graph is shown below in Figure 8. Proof. It is straightforward to verify that the rank of the adjacency matrix of the Bidiakis cube is eight, so we know that M(BC) ≥ . As a zero forcing set consider S = { , , , } with the vertex labeling in Figure 8. The Tutte-Coxeter graph TC is de ned to have a set of vertices denoted by   a , a , . . . , a , b , b , . . . , b , c , c , . . . where i ranges over { , , . . . , } and arithmetic is taken modulo . This graph is shown below in Figure 9.
-The Bidiakis Cube (cubic).  We also provide expansion techniques for constructing graphs with M(G) = Z(G) from smaller graphs with M(G) = Z(G) in Section 3.