A sharp upper bound on the spectral radius of θ(1, 3, 3)-free graphs with given size

A graph G is F-free if G does not contain F as a subgraph. Let ρ(G) be the spectral radius of a graph G. Let θ(1, p, q) denote the theta graph, which is obtained by connecting two distinct vertices with three internally disjoint paths with lengths 1, p, q, where p ≤ q. Let S_n,k denote the graph obtained by joining every vertex of K_k to n − k isolated vertices and S(formula Presented) denote the graph obtained from Sn,k by deleting an edge incident to a vertex of degree k, respectively. In this paper, we show that if (formula Presented) for a graph G with even size m ≥ 92, then G contains a θ(1, 3, 3) unless (formula Presented).