Laplacian spectral characterization of roses
Combinatorics
2017-09-20 v2
Abstract
A rose graph is a graph consisting of cycles that all meet in one vertex. We show that except for two specific examples, these rose graphs are determined by the Laplacian spectrum, thus proving a conjecture posed by Lui and Huang [F.J. Liu and Q.X. Huang, Laplacian spectral characterization of 3-rose graphs, Linear Algebra Appl. 439 (2013), 2914--2920]. We also show that if two rose graphs have a so-called universal Laplacian matrix with the same spectrum, then they must be isomorphic. In memory of Horst Sachs (1927-2016), we show the specific case of the latter result for the adjacency matrix by using Sachs' theorem and a new result on the number of matchings in the disjoint union of paths.
Keywords
Cite
@article{arxiv.1607.05466,
title = {Laplacian spectral characterization of roses},
author = {Changxiang He and Edwin R. van Dam},
journal= {arXiv preprint arXiv:1607.05466},
year = {2017}
}