Extremal spectral results of planar graphs without Cl, l or Theta graph

Let F be a given family of graphs. A graph G is F-free if it does not contain any member of F as a subgraph. Let Cl, l be a graph obtained from 2Cₗ such that the two cycles share a common vertex, where l3. A Theta graph is obtained from a cycle Cₖ by adding an additional edge between two non-consecutive vertices on Cₖ, where k 4. Let Θₖ be set of Theta graphs, where k 4. For sufficiently large n, the unique extremal planar graph with the maximum spectral radius among Cl, l-free planar graphs on n vertices and among Θₖ-free planar graphs on n vertices are characterized respectively, where l 3 and k 4.