Let $H_{s,t_1,\ldots ,t_k}$ be the graph with $s$ triangles and $k$ odd cycles of lengths $t_1,\ldots ,t_k\ge 5$ intersecting in exactly one common vertex. Recently, Hou, Qiu and Liu [Discrete Math. 341 (2018) 126--137], and Yuan [J. Graph Theory 89 (1) (2018) 26--39] determined independently the maximum number of edges in an $n$-vertex graph that does not contain $H_{s,t_1,\ldots ,t_k}$ as a subgraph. In this paper, we determine the graphs of order $n$ that attain the maximum spectral radius among all graphs containing no $H_{s,t_1,\ldots ,t_k}$ for $n$ large enough.