Constructing cospectral signed graphs
TLDR
In this paper, the Godsil-McKay algorithm was ported to signed graphs, and it was shown that with suitable adaption, such algorithms can be successfully ported to cospectral switching nonisomorphic signed graphs.Abstract:
A well--known fact in Spectral Graph Theory is the existence of pairs of isospectral nonisomorphic graphs (known as PINGS). The work of A.J. Schwenk (in 1973) and of C. Godsil and B. McKay (in 1982) shed some light on the explanation of the presence of isospectral graphs, and they gave routines to construct PINGS. Here, we consider the Godsil-McKay--type routines developed for graphs, whose adjacency matrices are $(0,1)$-matrices, to the level of signed graphs, whose adjacency matrices allow the presence of $-1$'s. We show that, with suitable adaption, such routines can be successfully ported to signed graphs, and we can build pairs of cospectral switching nonisomorphic signed graphs.read more
Citations
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Book ChapterDOI
An Introduction to the Theory of Graph Spectra: Spectral techniques
TL;DR: In this article, the authors introduce the concept of graph operations and modifications, and characterizations of spectra by characterizations by spectra and one eigenvalue, and Laplacians.
Journal ArticleDOI
Godsil-McKay switching for mixed and gain graphs over the circle group
TL;DR: In this article, the authors describe two methods, both inspired from Godsil-McKay switching on simple graphs, to build cospectral gain graphs whose gain group consists of the complex numbers of modulus 1 (the circle group).
Posted Content
Enumeration of Cospectral Graphs
Willem H. Haemers,Edward Spence +1 more
TL;DR: In this paper, the authors enumerated all graphs on at most 11 vertices and determined their spectra with respect to various matrices, such as the adjacency matrix and the Laplacian matrix.
Posted Content
Spectral Fundamentals and Characterizations of Signed Directed Graphs
Pepijn Wissing,Edwin van Dam +1 more
TL;DR: It is shown that non-empty signed directed graphs whose spectra occur uniquely, up to isomorphism, do not exist, but several infinite families whose spectRA occur uniquely up to (diagonal) switching equivalence are provided.
Journal ArticleDOI
Spectral fundamentals and characterizations of signed directed graphs
TL;DR: The spectral properties of signed directed graphs have received substantially less attention than those of their undirected and/or unsigned counterparts as discussed by the authors , which may be naturally obtained by assigning a sign to each edge of a directed graph.
References
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Book
Algebraic Graph Theory
Chris Godsil,Gordon F. Royle +1 more
TL;DR: The Laplacian of a Graph and Cuts and Flows are compared to the Rank Polynomial.
BookDOI
Spectra of graphs
TL;DR: This book gives an elementary treatment of the basic material about graph Spectra, both for ordinary, and Laplace and Seidel spectra, by covering standard topics before presenting some new material on trees, strongly regular graphs, two-graphs, association schemes, p-ranks of configurations and similar topics.
Book
Spectra of graphs : theory and application
TL;DR: The Spectrum and the Group of Automorphisms as discussed by the authors have been used extensively in Graph Spectra Techniques in Graph Theory and Combinatory Applications in Chemistry an Physics. But they have not yet been applied to Graph Spectral Biblgraphy.
Book ChapterDOI
An Introduction to the Theory of Graph Spectra: Spectral techniques
TL;DR: In this article, the authors introduce the concept of graph operations and modifications, and characterizations of spectra by characterizations by spectra and one eigenvalue, and Laplacians.
Journal ArticleDOI
Constructing cospectral graphs
TL;DR: In this article, some new constructions for families of cospectral graphs are derived, and some old constructions are considerably generalized, and one of these constructions is sufficiently powerful to produce an estimated 72% of the 51039 graphs on 9 vertices which do not have unique spectrum.
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