文献类型：期刊文献

英文题名：On the spectral extremal problem of planar graphs

作者：Wang, Xiaolong[1];Huang, Xueyi[1];Lin, Huiqiu[1]

机构：[1]East China Univ Sci & Technol, Sch Math, Shanghai 200237, Peoples R China

年份：2025

卷号：44

期号：8

外文期刊名：COMPUTATIONAL & APPLIED MATHEMATICS

收录：;EI(收录号：20253419022844);WOS:【SCI-EXPANDED(收录号:WOS:001554354700016)】；

基金：The authors are grateful to the reviewers for their valuable comments and helpful suggestions. X. Huang was supported by the National Natural Science Foundation of China (no. 12471324) and the Natural Science Foundation of Shanghai (no. 24ZR1415500). Huiqiu Lin was supported by the National Natural Science Foundation of China (no. 12271162), the Natural Science Foundation of Shanghai (nos. 22ZR1416300 and 23JC1401500) and the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning (no. TP2022031).

语种：英文

外文关键词：Spectral radius; Planar graph; Wheel graph; Friendship graph

摘要：The spectral extremal problem of planar graphs has received increasing attention in the past several decades. Boots and Royle (Geogr Anal 23(3):276-282, 1991) and Cao and Vince (Linear Algebra Appl 187:251-257, 1993 independently) conjectured that K2+Pn-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_2 + P_{n-2}$$\end{document} is the unique graph attaining the maximum spectral radius among all planar graphs on n vertices, where K2+Pn-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_2 + P_{n-2}$$\end{document} is the graph obtained from K2 boolean OR Pn-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_2\cup P_{n-2}$$\end{document} by adding all possible edges between K2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_2$$\end{document} and Pn-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{n-2}$$\end{document}. Tait and Tobin (J Combin Theory Ser B 126:37-161, 2017) confirmed this conjecture for all sufficiently large n. In this paper, we consider the spectral extremal problem for planar graphs without specified subgraphs. For a fixed graph F, let SPEXP(n,F)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{SPEX}_{{\mathcal {P}}}(n,F)$$\end{document} denote the set of graphs attaining the maximum spectral radius among all F-free planar graphs on n vertices. We describe a rough structure of the connected extremal graphs in SPEXP(n,F)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{SPEX}_{{\mathcal {P}}}(n,F)$$\end{document} when F is a planar graph not contained in K2,n-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{2,n-2}$$\end{document}. As applications, we determine the extremal graphs in SPEXP(n,Wk)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{SPEX}_{{\mathcal {P}}}(n,W_k)$$\end{document}, SPEXP(n,Fk)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{SPEX}_{{\mathcal {P}}}(n,F_k)$$\end{document} and SPEXP(n,Mk+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{SPEX}_{{\mathcal {P}}}(n,M_{k+1})$$\end{document} for all sufficiently large n, where Wk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_k$$\end{document}, Fk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_k$$\end{document} and Mk+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{k+1}$$\end{document} are the wheel graph of order k, the friendship graph of order 2k+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2k+1$$\end{document} and the (k+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(k+1)$$\end{document}-matching of order 2k+2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2k+2$$\end{document}, respectively.