Extremal spectral results of planar graphs without vertex-disjoint cycles

Given a planar graph family F ${\rm{ {\mathcal F} }}$, let exP(n,F) $e{x}_{{\mathscr{P}}}(n,{\mathscr{F}})$ and spexP(n,F) $spe{x}_{{\mathscr{P}}}(n,{\mathscr{F}})$ be the maximum size and maximum spectral radius over all n $n$-vertex F ${\rm{ {\mathcal F} }}$-free planar graphs, respectively. Let tCl $t{C}_{\ell }$ be the disjoint union of t $t$ copies of l $\ell $-cycles, and tC $t{\mathscr{C}}$ be the family of t $t$ vertex-disjoint cycles without length restriction. Tait and Tobin determined that K2+Pn-2 ${K}_{2}+{P}_{n-2}$ is the extremal spectral graph among all planar graphs with sufficiently large order n $n$, which implies the extremal graphs of both spexP(n,tCl) $spe{x}_{{\mathscr{P}}}(n,t{C}_{\ell })$ and spexP(n,tC) $spe{x}_{{\mathscr{P}}}(n,t{\mathscr{C}})$ for t >= 3 $t\ge 3$ are K2+Pn-2 ${K}_{2}+{P}_{n-2}$. In this paper, we first determine spexP(n,tCl) $spe{x}_{{\mathscr{P}}}(n,t{C}_{\ell })$ and spexP(n,tC) $spe{x}_{{\mathscr{P}}}(n,t{\mathscr{C}})$ and characterize the unique extremal graph for 1 <= t <= 2 $1\le t\le 2$, l >= 3 $\ell \ge 3$ and sufficiently large n $n$. Second, we obtain the exact values of exP(n,2C4) $e{x}_{{\mathscr{P}}}(n,2{C}_{4})$ and exP(n,2C) $e{x}_{{\mathscr{P}}}(n,2{\mathscr{C}})$, which solve a conjecture of Li for n >= 2661 $n\ge 2661$.