Extremal spectral results of planar graphs without $C_{l,l}$ or $\mathrm{Theta}$ graph

Abstract

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