Maximum Nullity Research Articles

A note on minimum rank and maximum nullity of sign patterns

The minimum rank of a sign pattern matrix is defined to be the smallest possible rank over all real matrices having the given sign pattern. The maximum nullity of a sign pattern is the largest possible nullity over the same set of matrices, and is equal to the number of columns minus the minimum rank of the sign pattern. Definitions of various graph parameters that have been used to bound maximum nullity of a zero-nonzero pattern, including path cover number and edit distance, are extended to sign patterns, and the SNS number is introduced to usefully generalize the triangle number to sign patterns. It is shown that for tree sign patterns (that need not be combinatorially symmetric), minimum rank is equal to SNS number, and maximum nullity, path cover number and edit distance are equal, providing a method to compute minimum rank for tree sign patterns. The minimum rank of small sign patterns is determined.

Minimum rank, maximum nullity and zero forcing number for selected graph families

The minimum rank of a simple graph G is dened to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i 6 j) is nonzero whenever fi;jg is an edge in G and is zero otherwise. Maximum nullity is taken over the same set of matrices, and the sum of maximum nullity and minimum rank is the order of the graph. The zero forcing number is the minimum size of a zero forcing set of vertices and bounds the maximum nullity from above. This paper denes the graph families ciclos and estrellas and establishes the minimum rank and zero forcing number of several of these families. In particular, these families provide the examples showing that the maximum nullity of a graph and its dual may dier, and similarly for zero forcing number.

Vertex and edge spread of zero forcing number, maximum nullity, and minimum rank of a graph

The minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i≠j) is nonzero whenever {i,j} is an edge in G and is zero otherwise; maximum nullity is taken over the same set of matrices. The zero forcing number is the minimum size of a zero forcing set of vertices and bounds the maximum nullity from above. The spread of a graph parameter at a vertex v or edge e of G is the difference between the value of the parameter on G and on G-v or G-e. Rank spread (at a vertex) was introduced in [4]. This paper introduces vertex spread of the zero forcing number and edge spreads for minimum rank/maximum nullity and zero forcing number. Properties of the spreads are established and used to determine values of the minimum rank/maximum nullity and zero forcing number for various types of grids with a vertex or edge deleted.

On minimum rank and zero forcing sets of a graph

For a graph G on n vertices and a field F , the minimum rank of G over F , written as mr F ( G ) , is the smallest possible rank over all n × n symmetric matrices over F whose ( i , j ) th entry (for i ≠ j ) is nonzero whenever ij is an edge in G and is zero otherwise. The maximum nullity of G over F is M F ( G ) = n - mr F ( G ) . The minimum rank problem of a graph G is to determine mr F ( G ) (or equivalently, M F ( G ) ). This problem has received considerable attention over the years. In [F. Barioli, W. Barrett, S. Butler, S.M. Cioabă, D. Cvetković, S.M. Fallat, C. Godsil, W. Haemers, L. Hogben, R. Mikkelson, S. Narayan, O. Pryporova, I. Sciriha, W. So, D. Stevanović, H. van der Holst, K.V. Meulen, A.W. Wehe, AIM Minimum Rank–Special Graphs Work Group, Zero forcing sets and the minimum rank of graphs, Linear Algebra Appl. 428 (2008) 1628–1648], a new graph parameter Z ( G ) , the zero forcing number, was introduced to bound M F ( G ) from above. The authors posted an attractive question: What is the class of graphs G for which Z ( G ) = M F ( G ) for some field F ? This paper focuses on exploring the above question.

Maximum generic nullity of a graph

For a graph G of order n , the maximum nullity of G is defined to be the largest possible nullity over all real symmetric n × n matrices A whose ( i , j ) th entry (for i ≠ j ) is nonzero whenever { i , j } is an edge in G and is zero otherwise. Maximum nullity and the related parameter minimum rank of the same set of matrices have been studied extensively. A new parameter, maximum generic nullity, is introduced. Maximum generic nullity provides insight into the structure of the null-space of a matrix realizing maximum nullity of a graph. It is shown that maximum generic nullity is bounded above by edge connectivity and below by vertex connectivity. Results on random graphs are used to show that as n goes to infinity almost all graphs have equal maximum generic nullity, vertex connectivity, edge connectivity, and minimum degree.

Maximum nullity of outerplanar graphs and the path cover number

Let G = ( V , E ) be a graph with V = { 1 , 2 , … , n } . Define S ( G ) as the set of all n × n real-valued symmetric matrices A = [ a ij ] with a ij ≠ 0 , i ≠ j if and only if ij ∈ E . By M ( G ) we denote the largest possible nullity of any matrix A ∈ S ( G ) . The path cover number of a graph G , denoted P ( G ) , is the minimum number of vertex disjoint paths occurring as induced subgraphs of G which cover all the vertices of G . There has been some success with relating the path cover number of a graph to its maximum nullity. Johnson and Duarte [5], have shown that for a tree T , M ( T ) = P ( T ) . Barioli et al. [2], show that for a unicyclic graph G , M ( G ) = P ( G ) or M ( G ) = P ( G ) - 1 . Notice that both families of graphs are outerplanar. We show that for any outerplanar graph G , M ( G ) ⩽ P ( G ) . Further we show that for any partial 2-path G , M ( G ) = P ( G ) .

Minimum rank problems

A graph describes the zero–nonzero pattern of a family of matrices, with the type of graph (undirected or directed, simple or allowing loops) determining what type of matrices (symmetric or not necessarily symmetric, diagonal entries free or constrained) are described by the graph. The minimum rank problem of the graph is to determine the minimum among the ranks of the matrices in this family; the determination of maximum nullity is equivalent. This problem has been solved for simple trees [P.M. Nylen, Minimum-rank matrices with prescribed graph, Linear Algebra Appl. 248 (1996) 303–316, C.R. Johnson, A. Leal Duarte, The maximum multiplicity of an eigenvalue in a matrix whose graph is a tree, Linear and Multilinear Algebra 46 (1999) 139–144], trees allowing loops [L.M. DeAlba, T.L. Hardy, I.R. Hentzel, L. Hogben, A. Wangsness. Minimum rank and maximum eigenvalue multiplicity of symmetric tree sign patterns, Linear Algebra Appl. 418 (2006) 389–415], and directed trees allowing loops [F. Barioli, S. Fallat, D. Hershkowitz, H.T. Hall, L. Hogben, H. van der Holst, B. Shader, On the minimum rank of not necessarily symmetric matrices: a preliminary study, Electron. J. Linear Algebra 18 (2000) 126–145]. We survey these results from a unified perspective and solve the minimum rank problem for simple directed trees.

Minimum rank of edge subdivisions of graphs

Let F be a field, let G be an undirected graph on n vertices, and let S(F, G)be the set of all F-valued symmetric n × n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. The minimum rank of G over F is defined to be mr(F, G) = min{rank A | A ∈ S(F, G)}. The problem of finding the minimum rank (maximum nullity) of edge subdivisions of a given graph G is investigated. Is is shown that if an edge is adjacent to a vertex of degree 1 or 2, its maximum nullity is unchanged upon subdividing the edge. This enables us to reduce the problem of finding the minimum rank of any graph obtained from G by subdividing edges to finding the minimum rank of those graphs obtained from G by subdividing each edge at most once. The graph obtained by subdividing each edge of G once is called its subdivision graph and is denoted by ❛G. It is shown that its maximum nullity is an upper bound for the maximum nullity of any graph obtained from G by subdividing edges. It is also shown that the minimum rank of ❛G often depends only upon the number of vertices of G. In conclusion, some illustrative examples and open questions are presented.

On the minimum rank of not necessarily symmetric matrices: A preliminary study

The minimum rank of a directed graph Γ is defined to be the smallest possible rank over all real matrices whose ijth entry is nonzero whenever (i,j) is an arc in Γ and is zero otherwise. The symmetric minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i ≠ j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. Maximum nullity is equal to the difference between the order of the graph and minimum rank in either case. Definitions of various graph parameters used to bound symmetric maximum nullity, including path cover number and zero forcing number, are extended to digraphs, and additional parameters related to minimum rank are introduced. It is shown that for directed trees, maximum nullity, path cover number, and zero forcing number are equal, providing a method to compute minimum rank for directed trees. It is shown that the minimum rank problem for any given digraph or zero-nonzero pattern may be converted into a symmetric minimum rank problem.

On the maximum positive semi-definite nullity and the cycle matroid of graphs

Let G =( V,E )b e ag raph withV = {1,2,...,n}, in which we allow parallel edges but no loops, and let S+(G) be the set of all positive semi-definite n × n matrices A =( ai,j )w ith ai,j =0i fij and i and j are non-adjacent, ai,jfij and i and j are connected by exactly one edge, and ai,j ∈ R if i = j or i and j are connected by parallel edges. The maximum positive semi-definite nullity of G, denoted by M+(G), is the maximum nullity attained by any matrix A ∈S +(G). A k-separation of G is a pair of subgraphs (G1,G2) such that V (G1) ∪ V (G2 )= V , E(G1) ∪ E(G2 )= E, E(G1) ∩ E(G2 )= ∅ and |V (G1) ∩ V (G2)| = k .W henG has a k-separation (G1,G2 )w ithk ≤ 2, we give a formula for the maximum positive semi-definite nullity of G in terms of G1,G2 ,a nd in case ofk = 2, also two other specified graphs. For a graph G ,l etcG denote the number of components in G. As a corollary of the result on k-separations with k ≤ 2, we obtain that M+(G) − cG = M+(G � ) − cG for graphs G and Gthat have isomorphic cycle matroids.

Three-connected graphs whose maximum nullity is at most three

For a graph G=(V,E) with vertex-set V={1,2,…,n}, let S(G) be the set of all n×n real-valued symmetric matrices A which represent G. The maximum nullity of a graph G, denoted by M(G), is the largest possible nullity of any matrix A∈S(G). Fiedler showed that a graph G has M(G)⩽1 if and only if G is a path. Johnson et al. gave a characterization of all graphs G with M(G)⩽2. Independently, Hogben and van der Holst gave a characterization of all 2-connected graphs with M(G)⩽2.In this paper, we show that k-connected graphs G have M(G)⩾k, that k-connected partial k-graphs G have M(G)=k, and that for 3-connected graphs G, M(G)⩽3 if and only if G is a partial 3-path.

On the nullity of bicyclic graphs

The nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum. In this paper, we obtain the nullity set of bicyclic graphs of order n, and determine the bicyclic graphs with maximum nullity.

On the nullity of graphs

The nullity of a graph G, denoted by η(G), is the multiplicity of the eigenvalue zero in its spectrum. It is known that η(G) ≤ n − 2 if G is a simple graph on n vertices and G is notisomorphic to nK1. In this paper, we characterize the extremal graphs attaining the upper bound n − 2 and the second upper bound n − 3. The maximum nullity of simple graphs with n vertices and e edges, M(n, e), is also discussed. We obtain an upper bound of M(n, e), and characterize n and e for which the upper bound is achieved.

Trees with maximum nullity

The nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum. Among n-vertex trees, the star has greatest nullity (equal to n − 2). We generalize this by showing that among n-vertex trees whose vertex degrees do not exceed a certain value D, the greatest nullity is n − 2⌈( n − 1)/ D⌉. Methods for constructing such maximum-nullity trees are described.

Graphs whose positive semi-definite matrices have nullity at most two

Let G=( V, E) be a undirected graph containing n vertices, and let M G be the set of all Hermitian n× n matrices M=( m i, j ) with m i, j ≠0 if i and j are connected by one edge of G, with m i,j∈ C if i and j are connected by at least two edges, with m i, j =0 if i≠ j, and i and j are not connected by an edge of G, and with m i, i for i=1,…, n a real number. What is the largest nullity attained by any positive semi-definite matrix M∈ M G ? In this paper we characterize, for t=1 and 2, those graphs G for which the maximum nullity is not greater than t.