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
## Explore with Wolfram|Alpha

## 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