Graphs whose minimal rank is two: The finite fields case

Barrett, Wayne; Holst, Hein van der; Loewy, Raphael

doi:10.13001/1081-3810.1175

Cited by 23 publications

(26 citation statements)

References 1 publication

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“…At the 2005 Oberwolfach graph theory workshop, Hein van der Holst asked if there are only finitely many k-critical graphs for any k and F . (Barrett, van der Holst, and Loewy [2,1] had recently confirmed this for k ≤ 2 and any F .) When F is finite, we answer this question affirmatively, by providing an upper bound on the size of a k-critical graph.…”

mentioning

confidence: 70%

On minimal rank over finite fields

Ding¹,

Kotlov²

2006

ELA

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Abstract. Let F be a field. Given a simple graph G on n vertices, its minimal rank (with respect to F ) is the minimum rank of a symmetric n × n F -valued matrix whose off-diagonal zeroes are the same as in the adjacency matrix of G. If F is finite, then for every k, it is shown that the set of graphs of minimal rank at most k is characterized by finitely many forbidden induced subgraphs, each on at most ( |F | k 2 + 1) 2 vertices. These findings also hold in a more general context.

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confidence: 70%

On minimal rank over finite fields

Ding¹,

Kotlov²

2006

ELA

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“…Barrett, van der Holst, and Loewy also have characterizations for infinite fields of characteristic two [5] and for finite fields [6]. These results provide tools that may allow characterizing minimum rank at most two over finite fields or fields of characteristic two.…”

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confidence: 89%

Minimum Rank of Graphs with Loops

Bozeman¹,

Ellsworth²,

Hogben

et al. 2014

ELA

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Abstract. A loop graph G is a finite undirected graph that allows loops but does not allow multiple edges. The set S(G) of real symmetric matrices associated with a loop graph G of order n is the set of symmetric matrices A = [a ij ] ∈ R n×n such that a ij = 0 if and only if ij ∈ E(G). The minimum (maximum) rank of a loop graph is the minimum (maximum) of the ranks of the matrices in S(G). Loop graphs having minimum rank at most two are characterized (by forbidden induced subgraphs and graph complements) and loop graphs having minimum rank equal to the order of the graph are characterized. A Schur complement reduction technique is used to determine the minimum ranks of cycles with various loop configurations; the minimum ranks of complete graphs and paths with various configurations of loops are also determined. Unlike simple graphs, loop graphs can have maximum rank less than the order of the graph. Some results are presented on maximum rank and which ranks between minimum and maximum can be realized. Interesting open questions remain.

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“…For example, Johnson and Leal Duarte [10] showed that the minimum rank of a tree equals the minimum number of disjoint paths needed to cover all vertices of the tree. Barrett, Loewy, and van der Holst [2,3] gave for any field F a combinatorial characterization of the class of graphs G with mr(G; F) ≤ 2. Also for the class of complement of trees, the minimum rank has been determined [8].…”

Section: Introductionmentioning

confidence: 99%

The inverse inertia problem for the complements of partial k -trees

Holst

2013

Linear Algebra and its Applications

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Let F be an infinite field with characteristic different from two. For a graph G = (V, E) with V = {1, . . . , n}, let S(G; F) be the set of all symmetric n × n matrices A = [a i,j ] over F with a i,j = 0, i = j if and only if ij ∈ E. We show that if G is the complement of a partial k-tree and m ≥ k + 2, then for all nonsingular symmetric m × m matrices K over F, there exists an m × n matrix U such that U T KU ∈ S(G; F). As a corollary we obtain that, if k + 2 ≤ m ≤ n and G is the complement of a partial k-tree, then for any two nonnegative integers p and q with p + q = m, there exists a matrix in S(G; R) with p positive and q negative eigenvalues.

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Graphs whose minimal rank is two: The finite fields case

Cited by 23 publications

References 1 publication

On minimal rank over finite fields

On minimal rank over finite fields

Minimum Rank of Graphs with Loops

The inverse inertia problem for the complements of partial k -trees

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