Denote by tCββ the disjoint union of t cycles of length β. Let
ex(n,F) and spex(n,F) be the maximum size and spectral radius over all
n-vertex F-free graphs, respectively. In this paper, we shall pay attention
to the study of both ex(n,tCββ) and spex(n,tCββ). On the one hand, we
determine ex(n,tC2β+1β) and characterize the extremal graph for any
integers t,β and nβ₯f(t,β), where f(t,β)=O(tβ2). This
generalizes the result on ex(n,tC3β) of Erd\H{o}s [Arch. Math. 13 (1962)
222--227] as well as the research on ex(n,C2β+1β) of F\"{u}redi and
Gunderson [Combin. Probab. Comput. 24 (2015) 641--645]. On the other hand, we
focus on the spectral Tur\'{a}n-type function spex(n,tCββ), and
determine the extremal graph for any fixed t,β and large enough n. Our
results not only extend some classic spectral extremal results on triangles,
quadrilaterals and general odd cycles due to Nikiforov, but also develop the
famous spectral even cycle conjecture proposed by Nikiforov (2010) and
confirmed by Cioab\u{a}, Desai and Tait (2022).Comment: 25 pages, one figur