The Maximum α-spectral Radius of Unicyclic Hypergraphs with Fixed Diameter

For 0 ≤ α < 1, the α-spectral radius of an r-uniform hypergraph G is the spectral radius of Aα(G)=αD(G)+(1−α)A(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\cal A}_\alpha}(G) = \alpha {\cal D}(G) + (1 - \alpha){\cal A}(G)$$\end{document}, where D(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal D}(G)$$\end{document} and A(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal A}(G)$$\end{document} are the diagonal tensor of degrees and adjacency tensor of G, respectively. In this paper, we show the perturbation of α-spectral radius by contracting an edge. Then we determine the unique unicyclic hypergraph with the maximum α-spectral radius among all r-uniform unicyclic hypergraphs with fixed diameter. We also determine the unique unicyclic hypergraph with the maximum α-spectral radius among all r-uniform unicyclic hypergraphs with given number of pendant edges.