Spectral extremal graphs for fan graphs

A well-known result of Nosal states that a graph $G$ with $m$ edges and $\lambda(G) > \sqrt{m}$ contains a triangle. Nikiforov [Combin. Probab. Comput. 11 (2002)] extended this result to cliques by showing that if $\lambda (G) > \sqrt{2m(1-1/r)}$, then $G$ contains a copy of $K_{r+1}$. Let $C_k^+$ be the graph obtained from a cycle $C_k$ by adding an edge to two vertices with distance two, and let $F_k$ 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 $m\ge 8$ and $\lambda (G) \ge \frac{1}{2} (1+\sqrt{4m-3})$, then $G$ contains a copy of $C_5,C_5^+$ and $F_2$, respectively, unless $G=K_2\vee \frac{m-1}{2}K_1$. In this paper, we give a unified extension by showing that such a graph contains a copy of $V_5$, where $V_5=K_1\vee P_4$ is the join of a vertex and a path on four vertices. Our result extends the aforementioned results since $C_5,C_5^+$ and $F_2$ are proper subgraphs of $V_5$. In addition, we prove that if $m\ge 33$ and $\lambda (G) \ge 1+ \sqrt{m-2}$, then $G$ contains a copy of $F_3$, unless $G=K_3\vee \frac{m-3}{3}K_1$. This confirms a conjecture on the friendship graph $F_k$ in the case $k=3$. Finally, we conclude some spectral extremal graph problems concerning the large fan graphs and wheel graphs.