A characterization of oriented hypergraphic balance via signed weak walks

Chen, Vinciane; Rao, Angeline; Rusnak, Lucas J.; Yang, Alex

doi:10.1016/j.laa.2015.08.001

Cited by 19 publications

(28 citation statements)

References 15 publications

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“…Moreover, if → P k maps into G via an incidence preserving map ω, then ω( → P k ) is determined by the sequence of (possibly repeating) incidences in G, hence is a weak walk in G. ◻ From here we are able to restate the weak walk Theorem for L G from [11,4] in terms of incidence preserving maps.…”

Section: Contributors As Generalized Basic Figuresmentioning

confidence: 74%

“…These definitions are an adaptation of those appearing in [12,11,4], and allow for oriented hypergraphs to be treated as locally signed graphic through its adjacencies.…”

Section: Oriented Hypergraphsmentioning

confidence: 99%

“…The (vertex) weak walk matrix of length k is the matrix W (G,V,V,k) whose (v, w)-entry is the number of positive weak walks of length k from v to w minus the number of negative weak walks of length k from v to w. It was shown in [11] for incidence-simple oriented hypergraphs, and improved to all oriented hypergraphs in [4], that the entries of oriented hypergraphic matrices are weak walk counts. 1.…”

Section: Oriented Hypergraphic Matricesmentioning

confidence: 99%

“…Theorems 4.1.1 and 4.2.1 allow for the study of optimizing permanents and determinants through contributors. The bounds presented are for integral incidence matrices -it may be possible to sharpen of the bounds via complex unit orientations by combining with the work from [4] and [10]. Figure 6 is extroverted with 6 contributors so perm(L G ) = 6.…”

Section: Optimizing Permanents and Determinantsmentioning

confidence: 99%

“…These theorems are unified and generalized by using the weak walk Theorem for oriented hypergraphs in [11,4], which unifies the entries of the oriented hypergraphic matrices as weak walk counts, then constructing incidence preserving maps from disjoint 1-paths into a given oriented hypergraph. Restrictions of these maps to adjacency preserving maps on sub-oriented-hypergraphs obtained by weak deletion of vertices allows for the reclaiming of basic figures of graphs, as well as the determinant of the adjacency matrix by cycle covers.…”

Section: Introductionmentioning

confidence: 99%

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A characterization of oriented hypergraphic Laplacian and adjacency matrix coefficients

Chen

Liu

Robinson

et al. 2018

Linear Algebra and its Applications

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An oriented hypergraph is an oriented incidence structure that generalizes and unifies graph and hypergraph theoretic results by examining its locally signed graphic substructure. In this paper we obtain a combinatorial characterization of the coefficients of the characteristic polynomials of oriented hypergraphic Laplacian and adjacency matrices via a signed hypergraphic generalization of basic figures of graphs. Additionally, we provide bounds on the determinant and permanent of the Laplacian matrix, characterize the oriented hypergraphs in which the upper bound is sharp, and demonstrate that the lower bound is never achieved. (Lucas J. Rusnak) 1 Portions of these results submitted to the 2016 Siemens Competition (regional semi-finalist). 2 Portions of these results appear in 2017 Master's Thesis.

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Section: Contributors As Generalized Basic Figuresmentioning

confidence: 74%

“…These definitions are an adaptation of those appearing in [12,11,4], and allow for oriented hypergraphs to be treated as locally signed graphic through its adjacencies.…”

Section: Oriented Hypergraphsmentioning

confidence: 99%

Section: Oriented Hypergraphic Matricesmentioning

confidence: 99%

Section: Optimizing Permanents and Determinantsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A characterization of oriented hypergraphic Laplacian and adjacency matrix coefficients

Chen

Liu

Robinson

et al. 2018

Linear Algebra and its Applications

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Graphs, Simplicial Complexes and Hypergraphs: Spectral Theory and Topology

Mulas

Horak²,

Jost

2022

Understanding Complex Systems

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Oriented hypergraphic matrix-tree type theorems and bidirected minors via Boolean order ideals

et al. 2018

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Restrictions of incidence preserving path maps produce oriented hypergraphic All Minors Matrix-tree Theorems for Laplacian and adjacency matrices. The images of these maps produce a locally signed graphic, incidence generalization, of cycle covers and basic figures that correspond to incidence-k-forests. When restricted to bidirected graphs, the natural partial ordering of maps results in disjoint signed Boolean lattices whose minor calculations correspond to principal order ideals. As an application, (1) the determinant formula of a signed graphic Laplacian is reclaimed and shown to be determined by the maximal positive-circle-free elements, and (2) spanning trees are equivalent to single-element order ideals.

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A characterization of oriented hypergraphic balance via signed weak walks

Cited by 19 publications

References 15 publications

A characterization of oriented hypergraphic Laplacian and adjacency matrix coefficients

A characterization of oriented hypergraphic Laplacian and adjacency matrix coefficients

Graphs, Simplicial Complexes and Hypergraphs: Spectral Theory and Topology

Oriented hypergraphic matrix-tree type theorems and bidirected minors via Boolean order ideals

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