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Limit points of eigenvalues of (di)graphs

Fu Ji Zhang, Zhibo Chen (2006)

Czechoslovak Mathematical Journal

The study on limit points of eigenvalues of undirected graphs was initiated by A. J. Hoffman in 1972. Now we extend the study to digraphs. We prove: 1. Every real number is a limit point of eigenvalues of graphs. Every complex number is a limit point of eigenvalues of digraphs. 2. For a digraph D , the set of limit points of eigenvalues of iterated subdivision digraphs of D is the unit circle in the complex plane if and only if D has a directed cycle. 3. Every limit point of eigenvalues of a set...

Line graphs: their maximum nullities and zero forcing numbers

Shaun Fallat, Abolghasem Soltani (2016)

Czechoslovak Mathematical Journal

The maximum nullity over a collection of matrices associated with a graph has been attracting the attention of numerous researchers for at least three decades. Along these lines various zero forcing parameters have been devised and utilized for bounding the maximum nullity. The maximum nullity and zero forcing number, and their positive counterparts, for general families of line graphs associated with graphs possessing a variety of specific properties are analysed. Building upon earlier work, where...

Main eigenvalues of real symmetric matrices with application to signed graphs

Zoran Stanić (2020)

Czechoslovak Mathematical Journal

An eigenvalue of a real symmetric matrix is called main if there is an associated eigenvector not orthogonal to the all-1 vector 𝐣 . Main eigenvalues are frequently considered in the framework of simple undirected graphs. In this study we generalize some results and then apply them to signed graphs.

Mapping directed networks.

Crofts, Jonathan J., Estrada, Ernesto, Higham, Desmond J., Taylor, Alan (2010)

ETNA. Electronic Transactions on Numerical Analysis [electronic only]

Matrix and discrepancy view of generalized random and quasirandom graphs

Marianna Bolla, Ahmed Elbanna (2016)

Special Matrices

We will discuss how graph based matrices are capable to find classification of the graph vertices with small within- and between-cluster discrepancies. The structural eigenvalues together with the corresponding spectral subspaces of the normalized modularity matrix are used to find a block-structure in the graph. The notions are extended to rectangular arrays of nonnegative entries and to directed graphs. We also investigate relations between spectral properties, multiway discrepancies, and degree...

Metric dimension and zero forcing number of two families of line graphs

Linda Eroh, Cong X. Kang, Eunjeong Yi (2014)

Mathematica Bohemica

Zero forcing number has recently become an interesting graph parameter studied in its own right since its introduction by the “AIM Minimum Rank–Special Graphs Work Group”, whereas metric dimension is a well-known graph parameter. We investigate the metric dimension and the zero forcing number of some line graphs by first determining the metric dimension and the zero forcing number of the line graphs of wheel graphs and the bouquet of circles. We prove that Z ( G ) ≤ 2 Z ( L ( G ) ) for a simple and connected graph G . Further,...

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