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# Laplacian Spectral Ratio

The Laplacian spectral ratio of a connected graph is defined as the ratio of its Laplacian spectral radius to its algebraic connectivity.

If a connected graph of even order satisfies , then has a perfect matching (Brouwer and Haemers 2005, Lin et al. 2023).

If is the maximum vertex degree and is the minimum vertex degree, then for a connected graph other than a complete graph,

(Goldberg 2006, Lin et al. 2023).

By the Kantorovich inequality, the Laplacian spectral ratio also satisfies the inequality

where is the Kirchhoff index and the edge count of a graph (Lin et al. 2023).

Algebraic Connectivity, Laplacian Matrix, Laplacian Polynomial, Laplacian Spectral Radius

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## References

Brouwer, A. E. and Haemers, W. H. "Eigenvalues and Perfect Matchings." Linear Algebra Appl. 395, 155-162, 2005.Goldberg, F. "Bounding the Gap Between Extremal Laplacian Eigenvalues of Graphs." Linear Algebra Appl. 416, 68-74, 2006.Haemers, W. H. "Interlacing Eigenvalues and Graphs." Linear Algebra Appl. 226-228, 593-616, 1995.Lin, Z.; Wang, J.; and Cai, M. "The Laplacian Spectral Ratio of Connected Graphs." 21 Feb 2023. https://arxiv.org/abs/2302.10491v1.

## Cite this as:

Weisstein, Eric W. "Laplacian Spectral Ratio." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LaplacianSpectralRatio.html