Ordinary and spectral extremal problems on vertex disjoint copies of even fans

Let $\mathrm{ex}(n, F)$ and $\mathrm{spex}(n, F)$ be the maximum size and spectral radius among all $F$-free graphs with fixed order $n$, respectively. A fan is a graph $P_1\vee P_{s}$ (join of a vertex and a path of order $s$) for $s\ge 3$, and it is called an even fan if $s$ is even. In this paper, we study $\mathrm{ex}(n,t(P_1\vee P_{2k}))$, $\mathrm{spex}(n,t(P_1\vee P_{2k}))$ with $t\ge 1$ and $k\ge 3$ and characterize the corresponding extremal graphs for sufficiently large $n$.