Laplacian spectral characterization of roses

TL;DR

This paper proves that most rose graphs are uniquely identified by their Laplacian spectra, confirming a prior conjecture, and extends spectral characterization to universal Laplacian matrices, with special cases for adjacency matrices.

Contribution

It establishes that all but two rose graphs are determined by their Laplacian spectrum and shows that identical universal Laplacian spectra imply graph isomorphism.

Findings

Most rose graphs are uniquely determined by their Laplacian spectrum.

Two rose graphs with the same universal Laplacian spectrum are isomorphic.

Specific case for adjacency matrix confirmed using Sachs' theorem and matchings in disjoint paths.

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.