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arXiv · 1607.05466

Laplacian spectral characterization of roses

Changxiang He·Edwin R. van Dam
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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.

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Changxiang He, Edwin R. van Dam. 2016-07-19. Laplacian spectral characterization of roses. https://doi.org/10.1016/j.laa.2017.08.012

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