Voltage Graphs

Groß, Joachim

doi:10.1201/b16132-48

Cited by 10 publications

(8 citation statements)

References 73 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As far as we know, one of the first papers where voltage graphs were used for construction of dense graphs is Alegre, Fiol and Yebra [1], but without using the name of 'voltage graphs'. This name was coined previously by Gross [11]. For more information, see Gross and Tucker [12], Baskoro, Branković, Miller, Plesník, Ryan and Siráň [3], and Miller and Siráň [14].…”

Section: Voltage and Lifted Digraphsmentioning

confidence: 99%

An algebraic approach to lifts of digraphs

Dalfó

Fiol

Miller

et al. 2019

Discrete Applied Mathematics

View full text Add to dashboard Cite

We study the relationship between two key concepts in the theory of (di)graphs: the quotient digraph, and the lift Γ α of a base (voltage) digraph. These techniques contract or expand a given digraph in order to study its characteristics, or obtain more involved structures. This study is carried out by introducing a quotient-like matrix, with complex polynomial entries, which fully represents Γ α . In particular, such a matrix gives the quotient matrix of a regular partition of Γ α , and when the involved group is Abelian, it completely determines the spectrum of Γ α . As some examples of our techniques, we study some basic properties of the Alegre digraph. In addition we completely characterize the spectrum of a new family of digraphs, which contains the generalized Petersen graphs, and that of the Hoffman-Singleton graph.

show abstract

Section: Voltage and Lifted Digraphsmentioning

confidence: 99%

An algebraic approach to lifts of digraphs

Dalfó

Fiol

Miller

et al. 2019

Discrete Applied Mathematics

View full text Add to dashboard Cite

show abstract

“…Using Gross 'voltage theory' (see [27]), in the bipartite case, and Stahl 'embedding schemes' (see [29]), in the non bipartite case, one can prove that given any cyclic permutation ε of ∆ d there exists a regular embedding i ε :Γ ֒→ F ε , where the surface F ε has euler characteristic…”

Section: Regular Genus Of Pl D-manifolds (Possibly With Boundary)mentioning

confidence: 99%

Lower bounds for regular genus and gem-complexity of PL 4-manifolds

Basak

Casali

2016

Forum Mathematicum

View full text Add to dashboard Cite

Within crystallization theory, two interesting PL invariants for d-manifolds have been introduced and studied, namely gem-complexity and regular genus. In the present paper we prove that, for any closed connected PL 4-manifold M , its gem-complexity k (M ) and its regular genus G(M ) satisfy:where rk(π 1 (M )) = m. These lower bounds enable to strictly improve previously known estimations for regular genus and gem-complexity of product 4-manifolds. Moreover, the class of semi-simple crystallizations is introduced, so that the represented PL 4-manifolds attain the above lower bounds. The additivity of both gem-complexity and regular genus with respect to connected sum is also proved for such a class of PL 4-manifolds, which comprehends all ones of "standard type", involved in existing crystallization catalogues, and their connected sums.MSC 2010 : Primary 57Q15. Secondary 57Q05, 57N13, 05C15.

show abstract

“…The concept of balance has been studied in the literature under various terminologies. For example, a balanced cycle is said to be satisfying Kirchhoff 's Voltage Law in [19]. In [9], the related concept to the signature of a cycle is the holonomy map.…”

Section: Frustration Index and Cheeger Constantsmentioning

confidence: 99%

Frustration index and Cheeger inequalities for discrete and continuous magnetic Laplacians

et al. 2015

View full text Add to dashboard Cite

We discuss a Cheeger constant as a mixture of the frustration index and the expansion rate, and prove the related Cheeger inequalities and higher order Cheeger inequalities for graph Laplacians with cyclic signatures, discrete magnetic Laplacians on finite graphs and magnetic Laplacians on closed Riemannian manifolds. In this process, we develop spectral clustering algorithms for partially oriented graphs and multi-way spectral clustering algorithms via metrics in lens spaces and complex projective spaces. As a byproduct, we give a unified viewpoint of Harary's structural balance theory of signed graphs and the gauge invariance of magnetic potentials.

show abstract

Voltage Graphs

Cited by 10 publications

References 73 publications

An algebraic approach to lifts of digraphs

An algebraic approach to lifts of digraphs

Lower bounds for regular genus and gem-complexity of PL 4-manifolds

Frustration index and Cheeger inequalities for discrete and continuous magnetic Laplacians

Contact Info

Product

Resources

About