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

Abstract

Let ${\text{spex}}_{\mathcal{P}}(n,F)$ and ${\text{spex}}_{\mathcal{O}\mathcal{P}}(n,F)$ 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 ${\text{spex}}_{\mathcal{P}}(n,B_{t,l})$ and ${\text{spex}}_{\mathcal{O}\mathcal{P}}(n,B_{t,l})$ 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 ${\text{spex}}_{\mathcal{P}}(n,C_{k,l})$ and ${\text{spex}}_{\mathcal{O}\mathcal{P}}(n,C_{k,l})$ along with their unique extremal graphs for all $l\ge k\ge 3$ and large $n$.

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