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Let Δ(T) and μ(T) denote the maximum degree and the Laplacian spectral radius of a tree T, respectively. In this paper we prove that for two trees T1 and T2 on n(n≥21) vertices, if Δ(T1)>Δ(T2) and Δ(T1)≥⌈11n30⌉+1, then μ(T1)>μ(T2), and the bound “Δ(T1)≥⌈11n30⌉+1” is the best possible. We also prove that for two trees T1 and T2 on 2k(k≥4) vertices with perfect matchings, if Δ(T1)>Δ(T2) and...
Let Δ(T) and μ(T) denote the maximum degree and the Laplacian spectral radius of a tree T, respectively. Let Tn be the set of trees on n vertices, and Tnc={T∈Tn∣Δ(T)=c}. In this paper, we determine the two trees which take the first two largest values of μ(T) of the trees T in Tnc when c≥⌈n2⌉. And among the trees in Tnc, the tree which alone minimizes the Laplacian spectral radius is characterized...
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