Spectral Extremal Problem on $t$ Copies of $\ell$-Cycles

Extremal problem on cycles plays an important role in extremal graph theory. 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 $ex(n,tC_\ell)$ and $spex(n,tC_\ell)$. On the one hand, we determine $ex(n,tC_{2\ell+1})$ and characterize the extremal graph for any integers $t,\ell$ and $n\geq f(t,\ell)$, where $f(t,\ell)=O(t\ell^2)$. This generalizes the result on $ex(n,tC_3)$ of Erdős [Arch. Math. 13 (1962) 222–227] as well as the research on $ex(n,C_{2\ell+1})$ of Füredi and Gunderson [Combin. Probab. Comput. 24 (2015) 641–645]. On the other hand, motivated by the spectral Turán-type problem proposed by Nikiforov, we obtain the extremal spectral radius $spex(n,tC_{\ell})$ for any fixed $t,\ell$ and large enough $n$. Our results extend some classic spectral extremal results or conjectures on odd cycles and even cycles. Our results also give some inspirations for general spectral Turán-type problem $spex(n,F)$ on bipartite or non-partite $F$.