Maximum nullity and zero forcing of circulant graphs

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 certain circulant graphs, including some bipartite circulants, cubic circulants, and circulants which are torus products, to obtain bounds on the minimum rank and the maximum nullity. We also evaluate when the zero forcing number will give equality.


Introduction
Let G be a simple nite graph with vertex set V(G) and edge set E(G). Suppose in the graph G some vertices are lled and some are un lled. The lling rule is as follows: if a vertex v ∈ V(G) is lled and has exactly one un lled neighbor, w, then vertex v forces w to be lled, and v is referred to as a forcing vertex. Given F ⊆ V(G), the nal lling of F is the set of lled vertices obtained by initially lling the vertices of F and leaving every vertex in V(G) \ F un lled and applying the lling rule until no more vertices can be lled. The set F is called a zero forcing set if the nal lling of F is V(G). The terminology of zero forcing arose in the context of forcing entries of a null vector to be zero as rst described in [2]. An example of a zero forcing set is given in Figure 1.
In various applications, it is of interest to nd the cardinality of a smallest zero forcing set in G (which always exists since V(G) is a trivial zero forcing set). The zero forcing number of G, denoted Z(G), is the minimum cardinality of a zero forcing set for a graph G. Determining Z(G) is NP-hard [1] in general, but has been calculated for some well-known classes of graphs (see, for example, [2,14,16]). Variations of zero forcing have been useful in communication complexity, quantum mechanics, electrical network monitoring, and G G Figure 1: The lled vertices are a zero forcing set in G but not in G .
some inverse eigenvalue problems (see [5,14] for references). The zero forcing number originated in [2] as a technique to nd a bound on the minimum rank of a symmetric matrix associated with a graph. Let S(G) denote the set of symmetric matrices over R whose graph is G. In particular, if G is a graph with vertices Recent work [3] describes families of graphs for which equality holds in Theorem 1.1, that is, families of graphs G with M(G) = Z(G). If we let mr(G) = min{rank(A) | A ∈ S(G)}, then the rank theorem tells us that mr(G) + M(G) = n. Hence Theorem 1.1 demonstrates that zero forcing can provide a lower bound on the minimum rank of any symmetric matrix associated with a graph.
Before going forward, we state some known facts about zero forcing, which can be found in [2]. In this paper we explore the zero forcing number for various classes of circulant graphs. Section 2 de nes circulant graphs and reviews some of their properties. In addition, we extend Deaett and Meyer's results on consecutive circulants [10]. The maximum nullity and zero forcing number of circulant graphs that are bipartite is explored in Section 3. For every bipartite circulant G considered in this section, Z(G) = M(G). Section 4 introduces the torus product of a graph, noting that certain circulant graphs can be viewed as a torus product. The section explores the zero forcing number and maximum nullity for several cases of torus products. The Möbius ladder is a special case of a torus product. We note that for many circulant graphs G which are torus products, the numbers Z(G) and M(G) are again equal, but there are still cases for which this is an open question. In Section 5, we show that all cubic circulant graphs G, Z(G) = M(G), and we compute this value.

Properties of circulant graphs
We recall some of the properties of circulant graphs, and derive some basic results on the zero forcing number and minimum rank for this family. For standard graph theory terminology, see [21]. Given an integer n ≥ and a subset S ⊆ { , , . . . , n }, a circulant graph G = Cn(S) is a graph with vertex set . . , n − } and j ∈ S}, taking sub-scripts modulo n. Note that if S = {s , . . . , s t }, then we will abuse notation and write Cn(s , . . . , s t ) instead of Cn({s , . . . , s t }). Furthermore, we assume that s < s < · · · < s t . Some examples of circulant graphs are given in Figures 2 and 3. Figure 2: The graph C ( , ) ∼ = K C (see Section 4) with a zero forcing set. Not every circulant graph is a connected graph (see Figure 4). The connected circulant graphs were characterized by Boesch and Tindell [6]; in the statement below, we write gG to denote g disjoint copies of the graph G.  Using the function f : Zn → Zn, de ned by f (x) = kx, Muzychuk [19] proved the following graph isomorphism between circulant graphs. As noted in the introduction, the zero forcing number (and minimum rank) of a number of families of graphs are known. Some of these families (e.g., complete graphs, cycles) are special cases of circulant graphs. The next theorem summarizes some of these known results. Given n ≥ and ≤ d ≤ n , the graphs Cn( , , . . . , d) are known as consecutive circulant graphs (e.g., see Figure 5). Deaett and Meyer [10] determined the zero forcing number and maximum nullity of consecutive circulants.

Families of bipartite circulants
In this section, we determine the zero forcing number and minimum rank of some families of bipartite circulant graphs using the work of Meyer [18]. Note that Meyer investigates the family of bipartite graphs whose biadjacency matrix is a circulant matrix. These graphs are sometimes called generalized bipartite circulants, although in [18], for expediency, they are simply called bipartite circulants. This usage is di erent than our usage of the term bipartite circulant graph. More precisely, a bipartite circulant (as used in this paper) is a circulant graph which is bipartite. In particular, the family of bipartite circulants is a subclass of the generalized bipartite circulants, the family of graphs studied in [18]. The reader should be aware of the two usages when consulting [18]. Our starting point is the following characterization of bipartite graphs due to Heuberger. Note that partitioning the vertices of a bipartite circulant into parts based on the parity of their index will provide a bipartition of the vertex set. Two bipartite circulant graphs are given in Figure 6.
, Figure 6: The bipartite circulant graphs C ( , ) ∼ = K , and C ( , ) Following Meyers [18], we can represent a bipartite circulant graph using its biadjacency matrix. Recall that if G is a biparitite graph with bipartition V ∪ V and m = |V | and n = |V |, then we can represent G by the m × n matrix A where A ij is if there is an edge between vertex v i ∈ V and vertex v j ∈ V , and 0 otherwise. The matrix A is the biadjacency matrix of G.
The next lemma describes how to represent the biadjacency matrix of a bipartite circulant graph. Below, P denotes the n × n permutation matrix corresponding to the n cycle ( · · · n). Note that P n = P = In and (1) If s t ≠ n, then the biadjacency matrix of G is (2) If s t = n, then the biadjacency matrix of G is Proof. This result is implicit in the proof of [18,Theorem 2.2]. In particular, it is shown that if s ∈ S, then s contributes the matrices P s− and P − s+ to the biadjacency matrix of C n (S) (there is a typo in [18] where the author has an n instead of an s). Note that P − s+ = P n− s+ . The result now follows by noting that if s t ≠ n, then each s i ∈ S gives two distinct matrices, but when s t = n, the two matrices P We recall two further results from Meyer's paper [18]; we have specialized his results to bipartite circulant graphs of the form C n (S).
Lemma 3.4. [18,Theorem 2.4] Suppose that G = C n (S) is a connected bipartite circulant graph with biadjacency matrix P i + · · · + P ir . Then for each unit a ∈ Z/nZ and element b ∈ Z/nZ, the graph G is isomorphic to the graph with biadjacency matrix P ai +b + · · · + P air+b where the exponents are computed modulo n.
We now come to the main results of this section.  (2), the biadjacency matrix of G has the form P k− + P k− + + · · · + P k + P k+ + · · · + P k+ . Let b = k − . Then by Lemma 3.4, G is isomorphic to the graph with biadjacency matrix P k− −b= + P + P + · · · + P k+ −b . Since k + − b = , by Theorem 3.5 we get M(G) = Z(G) = .
For our last result, we require the following result about complete bipartite graphs. (1) By Lemma 3.2, the biadjacency matrix of G has the form P + P + · · · + P − + P n− + · · · + P n− . Adding to each exponent, by Lemma 3.4, the graph G is isomorphic to the graph with the biadjacency matrix P + P + · · · + P − + . Employing Theorem 3.5 gives us M(G) = Z(G) = ( − ) = − .

Circulants which are torus products
In this section we extend the de nition of the Möbius ladder to a type of torus product. The zero forcing number for the Möbius ladder was calculated in [2] to be four (e.g., see Lemma 1.2 (7)). We compute the zero forcing number for our torus products, and as a corollary, we are able to compute the zero forcing number for a new family of circulant graphs. We also give evidence for a conjecture on the minimum rank of this family.
Recall that the Cartesian product of the graphs G and H with , (x k , y ) adjacent if either i = k and y j is adjacent to y in H, or j = and x i is adjacent to x k in G. We position the vertices of G H in a n × m grid such that the i-th column contains the vertices (x k , y i ), for ≤ k ≤ n, and the j-th row contains the vertices (x j , y k ), for ≤ k ≤ m. Then G H essentially consists of m copies of G as columns and n copies of H as rows. The product Cn Cm can be pictured as a lattice on a torus (see [17]). For m ≥ , de ne the torus product graph G Cm to consist of m copies, G , . . . Gm, of G with G i having vertices x ,i , x ,i , . . . , x n,i with edges between copies as follows: for ≤ i ≤ m − and ≤ k ≤ n, x k,i is adjacent to x k,i+ and, with subscript addition modulo n, x i,m is adjacent to x i+ , . Then the Möbius ladder is simply the torus product K Cm (see for example Figures 7 and 8). Note that the torus product Cn Cm is referred to as a twisted torus in [17].
The following proof takes advantage of the fact that the torus product Kn Cm is locally similar to the Cartesian product Kn Cm. Let G = Kn Cm for m ≥ . It was shown in [2] that Z(G) = M(G) = n. Below we give an alternative argument that Z(G) = n; the same argument applies to the torus product Kn Cm. Proof. Let G = Kn Cm or G = Kn Cm.
First consider the case m ≥ . Assume the vertices are in a grid as described before the theorem. The argument uses the fact that locally, about a column of vertices, the graphs of Kn Cm and Kn Cm both have the subgraph structure Kn P . (In fact there is an automorphism of G that takes column G i to G k for any i, k.) Observe that if n vertices of two adjacent copies of Kn are lled, this set is a zero forcing set of G. Thus, |Z(G)| ≤ n.
Let F be a minimum zero forcing set for G. Pick a forcing vertex v ∈ F. All but one neighbour of v must be in F. Since G is (n+ )-regular, |F| ≥ n+ . Once a forcing is made from v, the lled vertices are all those vertices in some copy of Kn (a column), plus two additional vertices in some copy of Cm (a row). These n + lled vertices are in three consecutive columns, say L, M, and R, the middle column M being completely lled.
In order for a vertex outside of these three columns to force some other vertex, it must be a lled vertex with at least n − lled neighbours. If these n vertices were originally in F, then |F| ≥ (n + ) + (n − ) = n. Thus, if |F| < n, then before a vertex outside of the three columns can do any forcing, some vertex in one of the three columns must rst force a vertex outside these columns.
Without loss of generality, let u be the rst vertex in column R used to force a vertex outside the three columns. Then the remaining vertices in R are already lled. Suppose r ≥ vertices of column R are in F. Then the remaining n − r vertices in column R must have been forced from vertices in column M. For any vertex in column M to force a vertex in column R, there must already be a vertex in column L, in the same row, that is lled. This implies that there must be at least n − r − vertices in column L that are in F and not adjacent to v. Hence |F| ≥ (n + ) + r + (n − r − ) = n. Therefore Z(G) = n. Now consider the case m = . Let F be a forcing set. As noted above, for a vertex v ∈ F to force another vertex, there must be at least (n − ) other vertices of F in the same column as v. As labelled above, M must

Remark 4.2.
As seen above, the proof takes advantage of the shared local structure of the Cartesian and torus products. Note the only di erence between the two graphs is the particular permutation of adjacencies between the rst column G and the last column Gn . As such, the same result holds true for a much larger class of graphs, if all of the n! permutations of the adjacencies between the rst and last column are considered, not just the two speci ed by G Cm and G Cm.
Theorem 4.1 can now be applied to the study of circulant graphs. In the next theorem we assume m ≥ since if m = then the graph considered is a consecutive circulant which is already discussed in Theorem 2.5. Proof. Let G = Cnm( , m, m, . . . , bm) with b = n . Then G can be obtained by combining the edges of Cnm( ) ∼ = Cnm and Cnm(m, m, . . . bm) ∼ = mCn( , , . . . , b) = mKn. One can then observe that for m ≥ , Cnm( , m, m, . . . , bm) is the torus product Kn Cm (see for example, Figures 8 and 9). The zero forcing number can then be obtained from Theorem 4.1.
If m = , then G is a (n + )-regular graph and by Lemma 1.2(1), Z(G) ≥ n + . Taking any vertex v and all but one of its neighbours provides a zero forcing set of size n + . In particular, v will force its only un lled neighbour. The remaining n − un lled vertices can be forced consecutively from the neighbours of v with subscripts that have the same parity as that of v.
We expect that M(G) = Z(G) for all the graphs in Theorem 4.3. Using special matrices, Theorem 4.5 and Theorem 4.7 demonstrate that M(G) = Z(G) for these graphs when m = and m = .
An n × n circulant Hankel matrix H is a matrix for which each row is shifted one position to the left from the row above it with a wrap around to the end of the row. In particular, if the rst row of H is (a , a , . . . , an), then the kth row of H is (a k , a k+ , . . . , a k− ). For example, is a circulant Hankel. Note that the reverse diagonals of a Hankel matrix are constant and consequently the matrix is symmetric. Let Note that if H is a circulant Hankel matrix, then PH = HP T and PH is itself a circulant Hankel matrix. Proof. Let H be a circulant Hankel matrix with rst row a = ( , , , . . . , n− , w) and w = − n− − . We claim that H has orthogonal rows. Since H is circulant, it is enough to show that the rst row of H is orthogonal to every other row of H. If b is row (k + ) of H, ≤ k ≤ n − , then b = ( k , k+ , . . . , n− , w, , , . . . , k− ) and Thus H = λI with λ = ||a|| = w + n− i= i . Therefore A = √ λ H is an orthogonal circulant Hankel matrix with no zero entries. The fact that A − PA has no zero entries follows from the fact that H ij ≠ H i+ ,j for ≤ i < n and ≤ j ≤ n.
Then, using the fact that AB = I − P T and BA = I − P, Since E is invertible, it follows that nulllity(K) ≥ n. Note that K is a symmetric matrix with graph Kn C , since A, B, PA and PB are symmetric matrices with no zero entries. Therefore, M(Kn C ) ≥ n and thus by Theorem 4.1, M(Kn C ) = n.
Since Kn C is the circulant C n ( , , , . . . , b) with b = n , we have the following:  I A I  I A  I  I PA  I  I  PA  I  P  I noting that APA = AAP T = AA T P T = P T . Since that rst and last n rows of EK are zero, and E is invertible, it follows that M(G) ≥ n and by Theorem 4.1, M(G) = n. There is another circulant graph that is isomorphic to a torus product. In particular, the graph Cnm( , m) ∼ = Cn Cm. (Note that for m ≠ n, Cm Cn is not isomorphic to Cn Cm.) The following theorem mimics a result [5] on the zero forcing number of the Cartesian product Cn Cm, which is not surprising since locally, the graph is the same as Cn Cm. In particular, it was shown in [5]  We do not know if M(G) = Z(G) in general for G = Cn Cm, but the zero forcing number is bounded above in same way as the Cartesian product, except when m = n and m is even. In this case, Z(Cm Cm) < Z(Cm Cm). The argument is similar to that in [11,Theorem 2.18]. If m is odd, then the forcing above will result in the two vertices in each of columns 1 and m being lled. In particular, rows k and k + will be completely lled, and so the resulting set will force the remaining rows to be lled, as noted at the beginning of the proof. If m is even, then column 1 will have the three vertices x k, , x (k+ ), , and x (k+ ), lled but column m will only have vertex x (k+ ),m lled. However, x (k+ ), can force x k,m . At this point, rows k and k + are completely lled, and so the remainder of the vertices can be forced. Therefore, Z(Cm Cm) ≤ m − .   Proof. The theorem follows from Theorem 5.1 and Theorems 5.2 and 5.3, along with Lemma 1.2 (5). Note that when a and m are odd, then the cubic circulant graphs C m (a, m) in Theorem 5.4 are further examples of bipartite circulants discussed in Section 3.

Concluding comment
For every circulant graph G for which we have calculated M(G) and Z(G), these two numbers have been equal. Equality also holds for the extreme cases; when G = Kn or G = Cn. We wonder if equality holds for every circulant graph in general.