Let \({\cal F}\) be a given family of graphs. A graph G is \({\cal F}\) -free if it does not contain any member of \({\cal F}\) as a subgraph. Let Cl,l be a graph obtained from 2Cl such that the two cycles share a common vertex, where l ⩾ 3. A Theta graph is obtained from a cycle Ck by adding an edge between two non-consecutive vertices on Ck, where k ⩾ 4. Let Θk be the set of Theta graphs on k vertices, where k ⩾ 4. For sufficiently large n, the extremal planar graphs with the maximum spectral radii among connected planar graphs on n vertices without Cl,l and among connected planar graphs on n vertices without Θk are characterized respectively, where l ⩾ 3 and k ⩾ 4.
