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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 a...
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... 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. ...
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... 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. ...
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.
... , n − 1 , for n ≥ 3 and k a positive integer less than ⌊ n 2 ⌋. In [2], the adjacency matrix was used to show that the maximum nullity is equal to the zero forcing number for certain generalized Petersen graphs. M(P (15r, 2)) = Z(P (15r, 2)) = 6 and M(P (24r, 5)) = Z(P (24r, 5)) = 12 and the maximum nullity is attained by the adjacency matrix. ...
... Applying two vertical and one horizontal subdivision edge insertion on the cube graph gives ECG(1,2). ...
It is known that the zero forcing number of a graph is an upper bound for the maximum nullity of the graph. 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 which bound the maximum nullity of a graph below. In particular, 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 Verdi\'ere type parameter and the 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.
... It was demonstrated in [2] that Z(G) provides an upperbound on M (G). 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. ...
... The consecutive circulants C 8 (1), C 8 (1, 2), C 8 (1, 2, 3), and C 8(1,2,3,4).By combining the above result with Lemma 2.3, one can determine Z(G) and M (G) for some other families of circulant graphs: Corollary 2.6. Suppose G = C n (s, 2s, 3s, . . . ...
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.
Given a graph G, one may ask: “What sets of eigenvalues are possible over all weighted adjacency matrices of G?” (The weight of an edge is positive or negative, while the diagonal entries can be any real numbers.) This is known as the Inverse Eigenvalue Problem for graphs (IEPG). A mild relaxation of this question considers the multiplicity list instead of the exact eigenvalues themselves. That is, given a graph G on n vertices and an ordered partition m=(m1,…,mℓ) of n, is there a weighted adjacency matrix where the i-th distinct eigenvalue has multiplicity mi? This is known as the ordered multiplicity IEPG. Recent work solved the ordered multiplicity IEPG for all graphs on 6 vertices.
In this work, we develop zero forcing methods for the ordered multiplicity IEPG in a multitude of different contexts. Namely, we utilize zero forcing parameters on powers of graphs to achieve bounds on consecutive multiplicities. We are able to provide general bounds on sums of multiplicities of eigenvalues for graphs. This includes new bounds on the sums of multiplicities of consecutive eigenvalues as well as more specific bounds for trees. Using these results, we verify the previous results above regarding the IEPG on six vertices. In addition, applying our techniques to skew-symmetric matrices, we are able to determine all possible ordered multiplicity lists for skew-symmetric matrices for connected graphs on five vertices.
Let $G$ be a simple undirected graph with each vertex colored either white or black, $ u $ be a black vertex of $ G, $ and exactly one neighbor $ v $ of $ u $ be white. Then change the color of $ v $ to black. When this rule is applied, we say $ u $ forces $ v, $ and write $ u \rightarrow v $. A $zero\ forcing\ set$ of a graph $ G$ is a subset $Z$ of vertices such that if initially the vertices in $ Z $ are colored black and remaining vertices are colored white, the entire graph $ G $ may be colored black by repeatedly applying the color-change rule. The zero forcing number of $ G$, denoted $Z(G), $ is the minimum size of a zero forcing set.\\ In this paper, we investigate the zero forcing number for the generalized Petersen graphs (It is denoted by $P(n,k)$). We obtain upper and lower bounds for the zero forcing number for $P(n,k)$. We show that $Z(P(n,2))=6$ for $n\geq 10$, $Z(P(n,3))=8$ for $n\geq 12$ and $Z(P(2k+1,k))=6$ for $k\geq 5$.
