The paper proves that extremal n-vertex graphs maximizing a feasible parameter P while forbidding C≥k, P_k or M_{k+1} belong to three explicit families of clique-join graphs, and derives applications to spectral and Turán problems.
Luo, The maximum number of cliques in graphs without l ong cycles, J
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Graph operations and a unified method for kinds of Tur\'an-type problems on paths, cycles and matchings
The paper proves that extremal n-vertex graphs maximizing a feasible parameter P while forbidding C≥k, P_k or M_{k+1} belong to three explicit families of clique-join graphs, and derives applications to spectral and Turán problems.