Spectral radius of graphs of given size with forbidden subgraphs

Let ρ(G) be the spectral radius of a graph G with m edges. Let S_m−k+1^k be the graph obtained from K_1,m−k by adding k disjoint edges within its independent set. Nosal's theorem states that if ρ(G)>m, then G contains a triangle. Zhai and Shu showed that any non-bipartite graph G with m≥26 and ρ(G)≥ρ(S_m¹)>m−1 contains a quadrilateral unless G≅S_m¹ M.Q. Zhai and J.L. Shu (2022) [25]. Wang proved that if ρ(G)≥m−1 for a graph G with size m≥27, then G contains a quadrilateral unless G is one out of four exceptional graphs Z.W. Wang (2022) [22]. In this paper, we show that any non-bipartite graph G with size m≥51 and ρ(G)≥ρ(S_m−1²)>m−2 contains a quadrilateral unless G≅S_m¹ or G≅S_m−1². Moreover, we show that if [Formula presented] for a graph G with even size m≥74, then G contains a C₅⁺ unless [Formula presented], where C_t⁺ denotes the graph obtained from C_t and C₃ by identifying an edge, S_n,k denotes the graph obtained by joining every vertex of K_k to n−k isolated vertices and S_n,k⁻ denotes the graph obtained from S_n,k by deleting an edge incident to a vertex of degree k, respectively.