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A well-known result of Nosal states that a graph G with m edges and (G) > m contains a triangle. Nikiforov Combin. Probab. Comput. 11 (2002) extended this result to cliques by showing that if (G) > 2m (1-1/r), then G contains a copy of Kₑ+₁. Let Cₖ^+ be the graph obtained from a cycle Cₖ by adding an edge to two vertices with distance two, and let Fₖ be the friendship graph consisting of k triangles that share a common vertex. Recently, Zhai, Lin and Shu European J. Combin. 95 (2021), Sun, Li and Wei Discrete Math. 346 (2023), and Li, Lu and Peng Discrete Math. 346 (2023) proved that if m 8 and (G) 12 (1+4m-3), then G contains a copy of C₅, C₅^+ and F₂, respectively, unless G=K₂ m-12K₁. In this paper, we give a unified extension by showing that such a graph contains a copy of V₅, where V₅=K₁ P₄ is the join of a vertex and a path on four vertices. Our result extends the aforementioned results since C₅, C₅^+ and F₂ are proper subgraphs of V₅. In addition, we prove that if m 33 and (G) 1+ m-2, then G contains a copy of F₃, unless G=K₃ m-33K₁. This confirms a conjecture on the friendship graph Fₖ in the case k=3. Finally, we conclude some spectral extremal graph problems concerning the large fan graphs and wheel graphs.
Yu et al. (Thu,) studied this question.