Abstract Let T be a tree with at least four vertices, none of which has degree 2, embedded in the plane. A Halin graph is a plane graph constructed by connecting the leaves of T into a cycle. Thus the cycle C forms the outer face of the Halin graph, with the tree inside it. Let G be a Halin graph with order n. Denote by μ ( G ) μ(G) the Laplacian spectral radius of G. This paper determines all the Halin graphs with μ ( G ) ≥ n − 4 μ(G)≥n−4 . Moreover, we obtain the graphs with the first three largest Laplacian spectral radius among all the Halin graphs on n vertices
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