Which graphs are determined by their spectrum?

Abstract

General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.

Eigenvalue bounds for the signless laplacian

“…The matrix D + A is called the signless Laplacian in [16], and it appears very rarely in published papers (see [5]), the paper [13] being almost the only relevant research paper published before 2003. Only recently has the signless Laplacian attracted the attention of researchers [4,8,11,10,16,26]. The present paper extends the survey [8] by providing further comments, proofs and conjectures.…”

“…Regular graphs can be recognized, and the degree of regularity and the number of components calculated, from Q G (λ), as noticed in [10]. In particular, we have the following proposition [8].…”

Some properties of the distance Laplacian eigenvalues of a graph

“…Note that these graphs are not cospectral with respect to the distance matrix, but they are with respect to the adjacency Laplacian. It is known [25] that the adjacency and (adjacency) Laplacian spectra are equivalent over the class of degree regular graphs. A similar result for the distance and the distance Laplacian spectra over the class of transmission regular graphs is proved in [3].…”

Lexicographic polynomials of graphs and their spectra

“…For example, it is known that the smallest pair of (connected) cospectral regular graphs have 10 vertices each [6]. So, we may use these graphs as a starting point to construct lexicographic polynomials that represent cospectral graphs.…”