200 research outputs found
Constructions of cospectral graphs with different zero forcing numbers
Several researchers have recently explored various graph parameters that can
or cannot be characterized by the spectrum of a matrix associated with a graph.
In this paper we show that several NP-hard zero forcing numbers are not
characterized by the spectra of several types of associated matrices with a
graph. In particular, we consider standard zero forcing, positive semidefinite
zero forcing, and skew zero forcing, and provide constructions of infinite
families of pairs of cospectral graphs which have different values for these
numbers. We explore several methods for obtaining these cospectral graphs
including using graph products, graph joins, and graph switching. Among these,
we provide a construction involving regular adjacency cospectral graphs; the
regularity of this construction also implies cospectrality with respect to
several other matrices including the Laplacian, signless Laplacian, and
normalized Laplacian. We also provide a construction where pairs of cospectral
graphs can have an arbitrarily large difference between their zero forcing
numbers
Developments on Spectral Characterizations of Graphs
In [E.R. van Dam and W.H. Haemers, Which graphs are determined by their spectrum?, Linear Algebra Appl. 373 (2003), 241-272] we gave a survey of answers to the question of which graphs are determined by the spectrum of some matrix associated to the graph. In particular, the usual adjacency matrix and the Laplacian matrix were addressed. Furthermore, we formulated some research questions on the topic. In the meantime some of these questions have been (partially) answered. In the present paper we give a survey of these and other developments.2000 Mathematics Subject Classification: 05C50Spectra of graphs;Cospectral graphs;Generalized adjacency matrices;Distance-regular graphs
New strongly regular graphs from finite geometries via switching
We show that the strongly regular graph on non-isotropic points of one type of the polar spaces of type U(n, 2), O(n, 3), O(n, 5), O+ (n, 3), and O- (n, 3) are not determined by its parameters for n >= 6. We prove this by using a variation of Godsil-McKay switching recently described by Wang, Qiu, and Hu. This also results in a new, shorter proof of a previous result of the first author which showed that the collinearity graph of a polar space is not determined by its spectrum. The same switching gives a linear algebra explanation for the construction of a large number of non-isomorphic designs. (C) 2019 Elsevier Inc. All rights reserved
Using twins and scaling to construct cospectral graphs for the normalized Laplacian
The spectrum of the normalized Laplacian matrix cannot determine the number
of edges in a graph, however finding constructions of cospectral graphs with
differing number of edges has been elusive. In this paper we use basic
properties of twins and scaling to show how to construct such graphs. We also
give examples of families of graphs which are cospectral with a subgraph for
the normalized Laplacian matrix
New Strongly Regular Graphs from Finite Geometries via Switching
We show that the strongly regular graph on non-isotropic points of one type
of the polar spaces of type , , , , and
are not determined by its parameters for . We prove this
by using a variation of Godsil-McKay switching recently described by Wang, Qiu,
and Hu. This also results in a new, shorter proof of a previous result of the
first author which showed that the collinearity graph of a polar space is not
determined by its spectrum. The same switching gives a linear algebra
explanation for the construction of a large number of non-isomorphic designs.Comment: 13 pages, accepted in Linear Algebra and Its Application
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