Minimizing the Laplacian spectral radius of trees with given matching number

Abstract

Let denote the set of trees on n vertices with fixed matching number β. In this article, we prove that if n = kβ +1, k ≥ 2, then the trees which minimize the Laplacian spectral radius over have maximum degree Δ =k, and determine the extremal trees for 1≤ β ≤4.

Keywords:

• Trees
• Laplacian eigenvalues
• Matching number

Keywords:

• AMS Classifications: 05C50
• 15A18

Acknowledgments

The authors are grateful to the referees for a short proof of in Theorem 3.3 and the outline of the proof of Theorem 3.6. They also thank the referees for their valuable comments, corrections, and suggestions which lead to a great improvement of this work.

This study was supported by National Natural Science Foundation of China (No. 10371075 and No. 10531070).