  The spectral extremal problem of planar graphs has aroused a lot of interest
over the past three decades. In 1991, Boots and Royle [Geogr. Anal. 23(3)
(1991) 276--282] (and Cao and Vince [Linear Algebra Appl. 187 (1993) 251--257]
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 $P_{n-2}$. In 2017, Tait and Tobin [J. Combin. Theory
Ser. B 126 (2017) 137--161] 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
$\mathrm{SPEX}_{\mathcal{P}}(n,F)$ denote the set of graphs attaining the
maximum spectral radius among all $F$-free planar graphs on $n$ vertices. We
describe a rough sturcture for the connected extremal graphs in
$\mathrm{SPEX}_{\mathcal{P}}(n,F)$ when $F$ is a planar graph not contained in
$K_{2,n-2}$. As applications, we determine the extremal graphs in
$\mathrm{SPEX}_{\mathcal{P}}(n,W_k)$, $\mathrm{SPEX}_{\mathcal{P}}(n,F_k)$ and
$\mathrm{SPEX}_{\mathcal{P}}(n,(k+1)K_2)$ for all sufficiently large $n$, where
$W_k$, $F_k$ and $(k+1)K_2$ are the wheel graph of order $k$, the friendship
graph of order $2k+1$ and the disjoint union of $k+1$ copies of $K_2$,
respectively.
