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arXiv · 1912.13227

Characterization of graphs with some normalized Laplacian eigenvalue of multiplicity n-3

Fenglei Tian·Dein Wong
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Abstract

Graphs with few distinct eigenvalues have been investigated extensively. In this paper, we focus on another relevant topic: characterizing graphs with some eigenvalue of large multiplicity. Specifically, the normalized Laplacian matrix of a graph is considered here. Let $\rho_{n-1}(G)$ and $\nu(G)$ be the second least normalized Laplacian eigenvalue and the independence number of a graph $G$, respectively. As the main conclusions, two families of $n$-vertex connected graphs with some normalized Laplacian eigenvalue of multiplicity $n-3$ are determined: graphs with $\rho_{n-1}(G)=-1$ and graphs with $\rho_{n-1}(G)\neq -1$ and $\nu(G)\neq 2$. Moreover, it is proved that these graphs are determined by their spectrum.

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Fenglei Tian, Dein Wong. 2019-12-31. Characterization of graphs with some normalized Laplacian eigenvalue of multiplicity n-3. https://arxiv.org/abs/1912.13227

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