Domination number and Laplacian eigenvalue of trees

Abstract

A dominating set in a graph G is a vertex set S⊆V(G) such that any vertex in V(G)\\S is adjacent to a vertex of S. The domination number γ(G) is the minimum cardinality of a dominating set of G. It is well known that γ(G)≥(d(G)+1)/3, where d(G) is the diameter of G. Let T be a tree on n vertices with diameter d(T). We show that γ(T)=(d(T)+1)/3 if and only if T contains a Laplacian eigenvalue of multiplicity n−d(T), and such trees are completely determined. As an application, we show that some trees are determined by their Laplacian spectra.