Denote by tC ℓ the disjoint union of t cycles of length ℓ. 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,tC ℓ ) and spex(n,tC ℓ ). On the one hand, we determine ex(n,tC 2ℓ+1 ) and characterize the extremal graph for any integers t, ℓ and n ≥ f (t, ℓ), where f (t, ℓ) = O(tℓ 2 ). This generalizes the result on ex(n,tC 3 ) of Erdős [Arch. Math. 13 (1962) 222-227] as well as the research on ex(n,C 2ℓ+1 ) of Füredi and Gunderson [Combin. Probab. Comput. 24 (2015) 641-645]. On the other hand, we focus on the spectral Turán-type function spex(n,tC ℓ ), and determine the extremal graph for any fixed t, ℓ 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 Cioabȃ, Desai and Tait (2022).
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