Abstract
For a graph G with domination number γ, Hedetniemi, Jacobs and Trevisan (2016) proved that , where
means the number of Laplacian eigenvalues of G in the interval
. Let T be a tree with diameter d. In this paper, we show that
. All trees achieving the lower bound are completely characterized. Moreover, we prove that the domination number of a tree is
if and only if it has exactly
Laplacian eigenvalues less than one. As an application, it also provides a new type of tree, which shows the sharpness of the inequality due to Hedetniemi, Jacobs and Trevisan.
2020 Mathematics Subject Classification:
Acknowledgments
The authors are grateful to the anonymous referees for checking the paper and providing constructive remarks.
Disclosure statement
No potential conflict of interest was reported by the author(s).