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Keywords:
Laplacian eigenvalue; ratio; tree; bound
Laplacian eigenvalue; ratio; tree; bound
Summary:
Let $G$ be a simple connected undirected graph. The Laplacian spectral ratio of $G$ is defined as the quotient between the largest and second smallest Laplacian eigenvalues of $G$, which is an important parameter in graph theory and networks. We obtain some bounds of the Laplacian spectral ratio in terms of the number of the spanning trees and the sum of powers of the Laplacian eigenvalues. In addition, we study the extremal Laplacian spectral ratio among trees with $n$ vertices, which improves some known results of Z. You and B. Liu (2012).
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