Spectral Gap of the Largest Eigenvalue of the Normalized Graph Laplacian

@article{Jost2019SpectralGO,
  title={Spectral Gap of the Largest Eigenvalue of the Normalized Graph Laplacian},
  author={Jurgen Jost and Raffaella Mulas and Florentin M{\"u}nch},
  journal={arXiv: Spectral Theory},
  year={2019}
}
We offer a new method for proving that the maximal eigenvalue of the normalized graph Laplacian of a graph with $n$ vertices is at least $\frac{n+1}{n-1}$ provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size $\frac{n-1}2$. With the same method, we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree, provided this is at most $\frac{n-1}{2… 
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Cheeger‐like inequalities for the largest eigenvalue of the graph Laplace operator
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A new Cheeger-like constant for graphs is defined and it is used for proving Cheager-like inequalities that bound the largest eigenvalue of the normalized Laplace operator.
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