2024/04/04 by Yu, Loujun, Li, Yongtao, Peng, Yuejian · 5 citations
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2404.03423
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\vee (m-1)/(2)K1. In this paper, we give a unified extension by showing that such a graph contains a copy of V5, where V5=K1\vee 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\vee (m-3)/(3)K1. 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.