Reducing the adjacency matrix of a tree

Fricke, Gerd H.; Hedetniemi, Stephen T.; Jacobs, Dean; Trevisan, Vilmar

doi:10.13001/1081-3810.1002

Cited by 19 publications

(9 citation statements)

References 6 publications

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“…But first let us generally relate maximum matchings of trees to eigenvectors of their null spaces. Maximum matchings of trees can be quite elegantly obtained using specialized algorithms [11], [20].…”

Section: Tree Eigenvector Decompositionmentioning

confidence: 99%

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Tree decomposition by eigenvectors

Sander

2009

Linear Algebra and its Applications

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In this work a composition-decomposition technique is presented that correlates tree eigenvectors with certain eigenvectors of an associated so-called skeleton forest. In particular, the matching properties of a skeleton determine the multiplicity of the corresponding tree eigenvalue. As an application a characterization of trees that admit eigenspace bases with entries only from the set {0, 1, −1} is presented. Moreover, a result due to Nylen concerned with partitioning eigenvectors of tree pattern matrices is generalized.

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Section: Tree Eigenvector Decompositionmentioning

confidence: 99%

“…That indeed a basis is formed follows from the fact that the rank of a tree equals twice the number of edges in a maximum matching of the tree (see e.g. [3], [11]).…”

Section: Linear Independence Of the Constructed Vectors Is Obvious Bymentioning

confidence: 99%

Tree decomposition by eigenvectors

Sander

2009

Linear Algebra and its Applications

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“…Then r(T ) = n if and only if T has a perfect matching. One can show that a tree has a perfect matching if and only if it is perfect (a proof of this result is Lemma 3.2 in [20]). Given that a graph is bipartite if and only if it has no cycles of odd length, a tree is bipartite since it is no cycles by definition.…”

Section: B Perfect Treesmentioning

confidence: 99%

Two-colorable graph states with maximal Schmidt measure

Severini

2006

Physics Letters A

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The Schmidt measure was introduced by Eisert and Briegel for quantifying the degree of entanglement of multipartite quantum systems [Phys. Rev. A 64, 022306 (2001)]. Although generally intractable, it turns out that there is a bound on the Schmidt measure for two-colorable graph states [Phys. Rev. A 69, 062311 (2004)]. For these states, the Schmidt measure is in fact directly related to the number of nonzero eigenvalues of the adjacency matrix of the associated graph. We remark that almost all two-colorable graph states have maximal Schmidt measure and we construct specific examples. These involve perfect trees, line graphs of trees, cographs, graphs from anti-Hadamard matrices, and unyciclic graphs. We consider some graph transformations, with the idea of transforming a two-colorable graph state with maximal Schmidt measure into another one with the same property. In particular, we consider a transformation introduced by François Jaeger, line graphs, and switching. By making appeal to a result of Ehrenfeucht et al. [Discrete Math. 278 (2004)], we point out that local complementation and switching form a transitive group acting on the set of all graph states of a given dimension.

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“…Indeed, there are algorithms to compute the determinant of the adjacency matrix of a tree in linear time [6] and the characteristic polynomial in time O(n 2 ) [7].…”

Section: Introductionmentioning

confidence: 99%

Efficient Computation of the Characteristic Polynomial of a Tree and Related Tasks

Fürer

2012

Algorithmica

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An O(n log 2 n) algorithm is presented to compute all coefficients of the characteristic polynomial of a tree on n vertices improving on the previously best quadratic time. With the same running time, the algorithm can be generalized in two directions. The algorithm is a counting algorithm for matchings, and the same ideas can be used to count other objects. For example, one can count the number of independent sets of all possible sizes simultaneously with the same running time. These counting algorithms not only work for trees, but can be extended to arbitrary graphs of bounded tree-width.

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Reducing the adjacency matrix of a tree

Cited by 19 publications

References 6 publications

Tree decomposition by eigenvectors

Tree decomposition by eigenvectors

Two-colorable graph states with maximal Schmidt measure

Efficient Computation of the Characteristic Polynomial of a Tree and Related Tasks

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