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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
25/2024

At the end of the spectrum: Chromatic bounds for the largest eigenvalue of the normalized Laplacian

Lies Beers and Raffaella Mulas

Abstract

For a graph with largest normalized Laplacian eigenvalue $\lambda_N$ and (vertex) coloring number $\chi$, it is known that $\lambda_N\geq \chi/(\chi-1)$. Here we prove properties of graphs for which this bound is sharp, and we study the multiplicity of $\chi/(\chi-1)$. We then describe a family of graphs with largest eigenvalue $\chi/(\chi-1)$. We also study the spectrum of the $1$-sum of two graphs, with a focus on the maximal eigenvalue. Finally, we give upper bounds on $\lambda_N$ in terms of $\chi$.

Received:
15.02.24
Published:
15.02.24

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Preprint
2024 Repository Open Access
Lies Beers and Raffaella Mulas

At the end of the spectrum : chromatic bounds for the largest eigenvalue of the normalized Laplacian