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).
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