Characterize trees with equality μ(T) = γ(T)

Characterize all finite trees T for which the number μ(T) of Laplacian eigenvalues of T in the interval [0,1), counted with multiplicity, equals the domination number γ(T); that is, determine precisely which trees satisfy μ(T) = γ(T).

Background

For a graph G, μ(G) denotes the count of Laplacian eigenvalues in [0,1), while γ(G) is the domination number. A general inequality μ(G) ≤ γ(G) holds for all graphs, and understanding when equality holds is a natural structural question.

The broader problem of characterizing all graphs with μ(G) = γ(G) was previously proposed in the literature. This paper resolves a separate question on bounding γ(T)/μ(T) for trees, but explicitly notes that the equality classification for trees remains open, highlighting its interest and relevance within the spectral–combinatorial interplay.