Spectral extremal graphs for fan graphs

A well-known result of Nosal states that a graph G with m edges and λ(G) > √m contains a triangle. Nikiforov [Combin. Probab. Comput. 11 (2002)] extended this result to cliques by showing that if λ (G) > √2m(1-1/r), then G contains a copy of Kᵣ₊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 λ (G) ≥ 1/2 (1+√4m-3), then G contains a copy of C₅,C₅⁺ and F₂, respectively, unless G=K₂ m-1/2K₁. 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 λ (G) ≥ 1+ √m-2, then G contains a copy of F₃, unless G=K₃ m-3/3K₁. 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.