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Laplacian Spectral Ratio

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

If a connected graph of even order satisfies R_L(G)>=2, then G has a perfect matching (Brouwer and Haemers 2005, Lin et al. 2023).

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

R_L>=(Delta+1)/delta

(Goldberg 2006, Lin et al. 2023).

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

K<=(n(n-1)^2)/(8m)(R_L+1/(R_L)+2),

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

See also

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