Extremal spectral results of planar graphs without vertex-disjoint cycles

Given a planar graph family $F$, let $e{x}_P(n,F)$ and $spe{x}_P(n,F)$ be the maximum size and maximum spectral radius over all $n$-vertex $F$-free planar graphs, respectively. Let $t{C}_{\ell }$ be the disjoint union of $t$ copies of $\ell$-cycles, and $tC$ be the family of $t$ vertex-disjoint cycles without length restriction. Tait and Tobin determined that $K_2+P_{n-2}$ is the extremal spectral graph among all planar graphs with sufficiently large order $n$, which implies the extremal graphs of both $spe{x}_P\left(n,t{C}_{\ell }\right)$ and $spe{x}_P\left(n,tC\right)$ for $t\ge 3$ are $K_2+P_{n-2}$. In this paper, we first determine $spe{x}_P\left(n,t{C}_{\ell }\right)$ and $spe{x}_P\left(n,tC\right)$ and characterize the unique extremal graph for $1\le t\le 2$, $\ell \ge 3$ and sufficiently large $n$. Second, we obtain the exact values of $e{x}_P(n,2C_4)$ and $e{x}_P(n,2C)$, which solve a conjecture of Li for $n\ge 2661$.