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 \(K_2 + P_{n-2}\) is the unique graph attaining the maximum spectral radius among all planar graphs on n vertices, where \(K_2 + P_{n-2}\) is the graph obtained from \(K_2\cup P_{n-2}\) by adding all possible edges between \(K_2\) and \(P_{n-2}\) . 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 \(\textrm{SPEX}_{{\mathcal {P}}}(n,F)\) 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 \(\textrm{SPEX}_{{\mathcal {P}}}(n,F)\) when F is a planar graph not contained in \(K_{2,n-2}\) . As applications, we determine the extremal graphs in \(\textrm{SPEX}_{{\mathcal {P}}}(n,W_k)\) , \(\textrm{SPEX}_{{\mathcal {P}}}(n,F_k)\) and \(\textrm{SPEX}_{{\mathcal {P}}}(n,M_{k+1})\) for all sufficiently large n, where \(W_k\) , \(F_k\) and \(M_{k+1}\) are the wheel graph of order k, the friendship graph of order \(2k+1\) and the \((k+1)\) -matching of order \(2k+2\) , respectively.
