Graph operations and a unified method for Turán-type problems on paths, cycles, and matchings

Abstract In this article, we present a unified approach for proving several Turán-type and generalized Turán-type problems, degree power problems, and extremal spectra problems on paths, cycles, and matchings. Specifically, we generalize classical results on cycles and matchings by Kopylov, Erdős–Gallai, and Luo et al., respectively, and provide a positive resolution to an open problem originally proposed by Nikiforov. Moreover, we improve the spectral extremal results on paths, building on the work of Nikiforov, and Nikiforov and Yuan. Additionally, we provide a comprehensive solution to the connected version of the problem related to the degree power sum of a graph that contains no path on k vertices, a topic initially investigated by Caro and Yuster.