A graph G is said to be F-free if it does not contain F as a subgraph. Let \(\mathcal {G}(m, F)\) denote the set of F-free graphs with m edges having no isolated vertices. A theta graph, denoted by \(\theta _{l_1,l_2,l_3}\) , is the graph obtained by connecting two distinct vertices with three internally disjoint paths of length \(l_1, l_2, l_3\) , where \(l_1\le l_2\le l_3\) and \(l_2\ge 2\) . Recently, Li, Zhao and Zou (2025) characterized the \(\theta _{1,p,q}\) -free graph of size m having the largest spectral radius, where \(q\ge p\ge 3\) and \(p+q\ge 2k+1\ge 7\) . Up to now, for all \(\theta _{1,p,q}\) -free graphs with \(q\ge p\ge 2\) , except for the case \(q=p=3\) , the graphs in \(\mathcal {G}(m,\theta _{1,p,q})\) with the largest spectral radius have been determined. So they proposed a problem on characterizing the graphs with the maximum spectral radius among \(\theta _{1,3,3}\) -free graphs. In this paper, we consider this problem and determine the maximum spectral radius of \(\theta _{1,3,3}\) -free graphs with size m and characterize the extremal graph.
