A relation between multiplicity of 1 as a Laplacian eigenvalue and induced matching numbers in trees

Abstract

The induced matching number of a graph is the largest number of edges at pairwise distance at least 2. Let T be a tree of order n with induced matching number ${\beta }^{\prime }\left(T\right)$. In this paper, we give a sharp upper bound in terms of n and ${\beta }^{\prime }\left(T\right)$ for the multiplicity of 1 as a Laplacian eigenvalue of T. Moreover, we characterize the extremal graphs.