ON THE NUMBER OF LAPLACIAN EIGENVALUES OF TREES SMALLER THAN TWO

Abstract

Let $m_T[0,2)$ be the number of Laplacian eigenvalues of a tree $T$ in $[0,2)$, multiplicities included. We give best possible upper bounds for $m_T[0,2)$ using the parameters such as the number of pendant vertices, diameter, matching number, and domination number, and characterize the trees $T$ of order $n$ with $m_T[0,2)=n-1$, $n-2$, and $\left \lceil \frac{n}{2} \right \rceil$, respectively, and in particular, show that $m_T[0,2)=\left \lceil \frac{n}{2} \right \rceil$ if and only if the matching number of $T$ is $\left \lfloor \frac{n}{2} \right \rfloor$.