A single-cone tree (unicyclic graph) is the join of a complete graph K1\documentclass[12pt]{minimal}
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\begin{document}$$K_1$$\end{document} and a tree (unicyclic graph). Suppose π=(d1,d2,…,dn)\documentclass[12pt]{minimal}
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\begin{document}$$\pi =(d_1, d_2, \ldots , d_n)$$\end{document} and π′=(d1′,d2′,…,dn′)\documentclass[12pt]{minimal}
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\begin{document}$$\pi ^{\,\prime }=(d_1^{\,\prime }, d_2^{\,\prime }, \ldots , d_n^{\,\prime })$$\end{document} are two non-increasing degree sequences. We say π\documentclass[12pt]{minimal}
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\begin{document}$$\pi $$\end{document} is majorizated by π′\documentclass[12pt]{minimal}
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\begin{document}$$\pi ^{\,\prime }$$\end{document}, denoted by π⊲π′\documentclass[12pt]{minimal}
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\begin{document}$$\pi \lhd \pi ^{\,\prime }$$\end{document}, if and only if π≠π′\documentclass[12pt]{minimal}
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\begin{document}$$\pi \ne \pi ^{\,\prime }$$\end{document}, ∑i=1ndi=∑i=1ndi′\documentclass[12pt]{minimal}
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\begin{document}$$\sum \nolimits _{i=1}^{n} d_i=\sum \nolimits _{i=1}^{n} d_i^{\,^{\,\prime }}$$\end{document}, and ∑i=1jdi≤∑i=1jdi′\documentclass[12pt]{minimal}
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\begin{document}$$\sum \nolimits _{i=1}^j d_i\le \sum \nolimits _{i=1}^j d_i^{\,^{\,\prime }}$$\end{document} for all j=1,2,…,n-1\documentclass[12pt]{minimal}
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\begin{document}$$j=1, 2, \ldots , n-1$$\end{document}. We use Jπ\documentclass[12pt]{minimal}
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\begin{document}$$J_{\pi }$$\end{document} to denote the class of single-cone trees (unicyclic graphs) with degree sequence π\documentclass[12pt]{minimal}
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\begin{document}$$\pi $$\end{document}. Suppose that π\documentclass[12pt]{minimal}
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\begin{document}$$\pi $$\end{document} and π′\documentclass[12pt]{minimal}
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\begin{document}$$\pi ^{\,\prime }$$\end{document} are two different non-increasing degree sequences of single-cone trees (unicyclic graphs). Let ρ\documentclass[12pt]{minimal}
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\begin{document}$$\rho $$\end{document} and ρ′\documentclass[12pt]{minimal}
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\begin{document}$$\rho ^{\,\prime }$$\end{document} be the largest spectral radius of the graphs in Jπ\documentclass[12pt]{minimal}
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\begin{document}$$J_{\pi }$$\end{document} and Jπ′\documentclass[12pt]{minimal}
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\begin{document}$$J_{\pi ^{\,\prime }}$$\end{document}, respectively, μ\documentclass[12pt]{minimal}
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\begin{document}$$\mu $$\end{document} and μ′\documentclass[12pt]{minimal}
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\begin{document}$$\mu ^{\,\prime }$$\end{document} be the largest signless Laplacian spectral radius of the graphs in Jπ\documentclass[12pt]{minimal}
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\begin{document}$$J_{\pi }$$\end{document} and Jπ′\documentclass[12pt]{minimal}
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\begin{document}$$J_{\pi ^{\,\prime }}$$\end{document}, respectively. In this paper, we prove that if π⊲π′\documentclass[12pt]{minimal}
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\begin{document}$$\pi \lhd \pi ^{\,\prime }$$\end{document}, then ρ<ρ′\documentclass[12pt]{minimal}
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\begin{document}$$\rho <\rho ^{\,\prime }$$\end{document} and μ<μ′\documentclass[12pt]{minimal}
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\begin{document}$$\mu <\mu ^{\,\prime }$$\end{document}.
