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37
The Minimum Rank of Symmetric Matrices Described by a Graph: A Survey
, 2007
"... 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. This paper surveys the current state of knowledge on the problem of determining the minim ..."
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Cited by 42 (18 self)
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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. This paper surveys the current state of knowledge on the problem of determining the minimum rank of a graph and related issues.
ThreeDimensional Orthogonal Graph Drawing
, 2000
"... vi Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . ..."
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Cited by 33 (13 self)
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vi Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii List of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv I Orthogonal Graph Drawing 1 1
Planarizing Graphs  A Survey and Annotated Bibliography
, 1999
"... Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph parameters. While there are many algorithmic results about planarization through edge deletion, the results abo ..."
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Cited by 32 (0 self)
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Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph parameters. While there are many algorithmic results about planarization through edge deletion, the results about vertex splitting, thickness, and crossing number are mostly of a structural nature. We also include a brief section on vertex deletion. We do not consider parallel algorithms, nor do we deal with online algorithms.
A Borsuk theorem for antipodal links and a spectral characterization of linklessly embeddable graphs
"... For any undirected graph G, let µ(G) be the graph parameter introduced by Colin de Verdière. In this paper we show that (G) 4 if and only if G is linklessly embeddable (in R 3 ). This forms a spectral characterization of linklessly embeddable graphs, and was conjectured by Robertson, Seymour, and ..."
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Cited by 29 (8 self)
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For any undirected graph G, let µ(G) be the graph parameter introduced by Colin de Verdière. In this paper we show that (G) 4 if and only if G is linklessly embeddable (in R 3 ). This forms a spectral characterization of linklessly embeddable graphs, and was conjectured by Robertson, Seymour, and Thomas. A key ingredient is a Borsuktype theorem on the existence of a pair of antipodal linked (k \Gamma 1) spheres in certain mappings OE : S 2k ! R 2k\Gamma1 . This result might be of interest in its own right. We also derive that (G) 4 for each linklessly embeddable graph G = (V; E), where (G) is the graph paramer introduced by van der Holst, Laurent, and Schrijver. (It is the largest dimension of any subspace L of R V such that for each nonzero x 2 L, the positive support of x induces a nonempty connected subgraph of G.)
The Colin de Verdière graph parameter
, 1997
"... In 1990, Y. Colin de Verdière introduced a new graph parameter (G), based on spectral properties of matrices associated with G. He showed that (G) is monotone under taking minors and that planarity of G is characterized by the inequality (G) 3. Recently Lovasz and Schrijver showed that linkless emb ..."
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Cited by 16 (1 self)
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In 1990, Y. Colin de Verdière introduced a new graph parameter (G), based on spectral properties of matrices associated with G. He showed that (G) is monotone under taking minors and that planarity of G is characterized by the inequality (G) 3. Recently Lovasz and Schrijver showed that linkless embeddability of G is characterized by the inequality (G) 4. In this paper we give an overview of results on (G) and of techniques to handle it.
A variant on the graph parameters of Colin de Verdière: Implications to the minimum rank of graphs
 Electronic Journal of Linear Algebra
"... Abstract. For a given undirected graph G, the minimum rank of G is defined to be the smallest possible rank over all real symmetric matrices A whose (i, j)th entry is nonzero whenever i 6 = j and {i, j} is an edge in G. Building upon recent work involving maximal coranks (or nullities) of certain sy ..."
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Cited by 14 (9 self)
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Abstract. For a given undirected graph G, the minimum rank of G is defined to be the smallest possible rank over all real symmetric matrices A whose (i, j)th entry is nonzero whenever i 6 = j and {i, j} is an edge in G. Building upon recent work involving maximal coranks (or nullities) of certain symmetric matrices associated with a graph, we introduce a new parameter ξ which is also based on the corank of a different but related class of symmetric matrices. For this new parameter we verify some properties analogous to the ones possessed by the existing parameters and, in addition, we attempt to apply these properties associated with ξ, to learn more about the minimum rank of graphs – our original motivation.
The Colin de Verdière number and sphere representations of a graph
, 1996
"... Colin de Verdi`ere introduced an interesting linear algebraic invariant (G) of graphs. He proved that (G) 2 if and only if G is outerplanar, and (G) 3 if and only if G is planar. We prove that if the complement of a graph G on n nodes is outerplanar, then (G) n \Gamma 4, and if it is planar, t ..."
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Cited by 13 (7 self)
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Colin de Verdi`ere introduced an interesting linear algebraic invariant (G) of graphs. He proved that (G) 2 if and only if G is outerplanar, and (G) 3 if and only if G is planar. We prove that if the complement of a graph G on n nodes is outerplanar, then (G) n \Gamma 4, and if it is planar, then (G) n \Gamma 5. We give a full characterization of maximal planar graphs whose complements G have (G) = n \Gamma 5. In the opposite direction we show that if G does not have "twin" nodes, then (G) n \Gamma 3 implies that the complement of G is outerplanar, and (G) n \Gamma 4 implies that the complement of G is planar. Our main tools are a geometric formulation of the invariant, and constructing representations of graphs by spheres, related to the classical result of Koebe about representing planar graphs by touching disks. In particular we show that such sphere representations characterize outerplanar and planar graphs.
Spectral graph theory and the inverse eigenvalue of a graph. Presentation at
 Directions in Combinatorial Matrix Theory Workshop, Banff International Research Station
, 2004
"... Abstract. Spectral Graph Theory is the study of the spectra of certain matrices defined from a given graph, including the adjacency matrix, the Laplacian matrix and other related matrices. Graph spectra have been studied extensively for more than fifty years. In the last fifteen years, interest has ..."
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Cited by 11 (2 self)
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Abstract. Spectral Graph Theory is the study of the spectra of certain matrices defined from a given graph, including the adjacency matrix, the Laplacian matrix and other related matrices. Graph spectra have been studied extensively for more than fifty years. In the last fifteen years, interest has developed in the study of generalized Laplacian matrices of a graph, that is, real symmetric matrices with negative offdiagonal entries in the positions described by the edges of the graph (and zero in every other offdiagonal position). The set of all real symmetric matrices having nonzero offdiagonal entries exactly where the graph G has edges is denoted by S(G). Given a graph G, the problem of characterizing the possible spectra of B, such that B ∈S(G), has been referred to as the Inverse Eigenvalue Problem of a Graph. In the last fifteen years a number of papers on this problem have appeared, primarily concerning trees. The adjacency matrix and Laplacian matrix of G and their normalized forms are all in S(G). Recent work on generalized Laplacians and Colin de Verdière matrices is bringing the two areas closer together. This paper surveys results in Spectral Graph Theory and the Inverse Eigenvalue Problem of a Graph, examines the connections between these problems, and presents some new results on construction of a matrix of minimum rank for a given graph having a special form such as a 0,1matrix or a generalized Laplacian.
Forbidden minors for the class of graphs G with ξ(G
"... Abstract. For a given simple graph G, S(G) is defined to be the set of real symmetric matrices A whose (i, j)th entry is nonzero whenever i = j and ij is an edge in G. In [2], ξ(G) is defined to be the maximum corank (i.e., nullity) among A ∈ S(G) having the Strong Arnold Property; ξ is used to stu ..."
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Cited by 11 (5 self)
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Abstract. For a given simple graph G, S(G) is defined to be the set of real symmetric matrices A whose (i, j)th entry is nonzero whenever i = j and ij is an edge in G. In [2], ξ(G) is defined to be the maximum corank (i.e., nullity) among A ∈ S(G) having the Strong Arnold Property; ξ is used to study the minimum rank/maximum eigenvalue multiplicity problem for G. Since ξ is minor monotone, the graphs G such that ξ(G) ≤ k can be described by a finite set of forbidden minors. We determine the forbidden minors for ξ(G) ≤ 2 and present an application of this characterization to computation of minimum rank among matrices in S(G).