Spectral extremal problem on \(t\) copies of \(\ell\)-cycles (Q6635172)

The paper addresses the spectral extremal problem for \(t\)-disjoint copies of \(\ell\)-cycles (\(tC_\ell\)) in the context of extremal graph theory. The authors investigate the Turán number (\(ex(n, tC_\ell)\)) and spectral extremal radius (\(spex(n, tC_\ell)\)) for graphs that are free of such configurations. This work extends classical results by \textit{P. Erdős} [Arch. Math. 13, 222--227 (1962; Zbl 0105.17504)], \textit{Z. Füredi} and \textit{D. S. Gunderson} [Comb. Probab. Comput. 24, No. 4, 641--645 (2015; Zbl 1371.05142)] to general odd and even cycles while contributing to the spectral Turán-type framework initiated by \textit{V. Nikiforov} [Linear Algebra Appl. 427, No. 2--3, 183--189 (2007; Zbl 1128.05035)]. The primary contributions include new characterizations of extremal graphs and explicit formulas for both \(ex(n, tC_\ell)\) and \(spex(n, tC_\ell)\) for large \(n\).\N\NThe methodology relies on advanced spectral graph theory tools, including Fourier transformation techniques, the Rayleigh quotient, and stability results derived from spectral extremal methods. The authors also utilize structural graph properties, such as maximum degree constraints and matching numbers, to rigorously prove their main theorems. These results offer a deeper understanding of the interplay between combinatorial and spectral properties in extremal graph theory. Moreover, the implications include insights into broader spectral extremal problems, inspiring further investigation into bipartite and non-bipartite configurations.