Spectral extremal problems on planar and outerplanar graphs without $C_{k,l}

Let $\emph{spex}_{\mathcal{P}}(n,F)$ and $\emph{spex}_{\mathcal{OP}}(n,F)$ be the maximum spectral radius among all $n$-vertex $F$-free planar graphs and outerplanar graphs, respectively.

Define $C_{k,l}$ as a graph obtained from $C_k \cup C_l$ such that the two cycles share a common vertex, where $l \ge k \ge 3$.

In the 1990s, Cvetković and Rowlinson conjectured $K_1 + P_{n-1}$ maximizes spectral radius in outerplanar graphs on $n$ vertices, while Boots and Royle (independently, Cao and Vince) conjectured $K_2 + P_{n-2} $ does so in planar graphs.

Tait and Tobin [J.

Combin.

Theory Ser.

B, 2017] determined the fundamental structure as the key to confirming these two conjectures for sufficiently large $n$.

Recently, Yin and Li [Discrete Mathematics, 2026] characterized the extremal graphs for $\emph{spex}_{\mathcal{P}}(n,B_{t,l})$ and $\emph{spex}_{\mathcal{OP}}(n,B_{t,l})$ in planar and outerplanar graphs on the basis of this key idea, where $B_{t,l}$ denotes the graph obtained by $t$ edge-disjoint $l$-cycles sharing a common vertex.

In this paper, we focus on planar and outerplanar graphs without $C_{k,l}$, and determine $\emph{spex}_{\mathcal{P}}(n,C_{k,l})$ and $\emph{spex}_{\mathcal{OP}}(n,C_{k,l})$ along with their unique extremal graphs for all $l \geq k \geq 3$ and large $n$.