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
TLDR
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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References

SHOWING 1-5 OF 5 REFERENCES
Bounds on normalized Laplacian eigenvalues of graphs
Let G be a simple connected graph of order n, where n≥2. Its normalized Laplacian eigenvalues are 0=λ1≤λ2≤⋯≤λn≤2. In this paper, some new upper and lower bounds on λn are obtained, respectively.…
On the spectrum of the normalized Laplacian for signed graphs: Interlacing, contraction, and replication
Extremal graph on normalized Laplacian spectral radius and energy
Let G = (V, E) be a simple graph of order n and the normalized Laplacian eigenvalues ρ1 ≥ ρ2 ≥ · · · ≥ ρn−1 ≥ ρn = 0. The normalized Laplacian energy (or Randić energy) of G without any isolated…
Spectral Graph Theory
Eigenvalues and the Laplacian of a graph Isoperimetric problems Diameters and eigenvalues Paths, flows, and routing Eigenvalues and quasi-randomness Expanders and explicit constructions Eigenvalues…
On graphs with at least three distance eigenvalues less than −1☆