Extremal spectral results of planar graphs without vertex-disjoint cycles

Given a planar graph family $\mathcal{F}$, let ${\rm ex}_{\mathcal{P}}(n,\mathcal{F})$ and ${\rm spex}_{\mathcal{P}}(n,\mathcal{F})$ be the maximum size and maximum spectral radius over all $n$-vertex $\mathcal{F}$-free planar graphs, respectively. Let $tC_{\ell}$ be the disjoint union of $t$ copies of $\ell$-cycles, and $t\mathcal{C}$ be the family of $t$ vertex-disjoint cycles without length restriction. Tait and Tobin [Three conjectures in extremal spectral graph theory, J. Combin. Theory Ser. B 126 (2017) 137--161] 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 ${\rm spex}_{\mathcal{P}}(n,tC_{\ell})$ and ${\rm spex}_{\mathcal{P}}(n,t\mathcal{C})$ for $t\geq 3$ are $K_2+P_{n-2}$. In this paper, we first determine ${\rm spex}_{\mathcal{P}}(n,tC_{\ell})$ and ${\rm spex}_{\mathcal{P}}(n,t\mathcal{C})$ and characterize the unique extremal graph for $1\leq t\leq 2$, $\ell\geq 3$ and sufficiently large $n$. Secondly, we obtain the exact values of ${\rm ex}_{\mathcal{P}}(n,2C_4)$ and ${\rm ex}_{\mathcal{P}}(n,2\mathcal{C})$, which solve a conjecture of Li [Planar Tur\'an number of the disjoint union of cycles, Discrete Appl. Math. 342 (2024) 260--274] for $n\geq 2661$.