Spectral extremal problems on outerplanar and planar graphs

Abstract

Let ${\text{spex}}_{\mathrm{OP}}(n,F)$ and ${\text{spex}}_P(n,F)$ be the maximum spectral radius over all n-vertex F-free outerplanar graphs and planar graphs, respectively. Define $t{C}_l$ as t vertex-disjoint l-cycles, $B_{t,l}$ as the graph obtained by sharing a common vertex among t edge-disjoint l-cycles and $(t+1)K_2$ as the disjoint union of $t+1$ copies of $K_2$. In the 1990s, Cvetković and Rowlinson conjectured $K_1\vee P_{n-1}$ maximizes spectral radius in outerplanar graphs on n vertices, while Boots and Royle (independently, Cao and Vince) conjectured $K_2\vee P_{n-2}$ does so in planar graphs. Tait and Tobin (2017) [22] determined the fundamental structure as the key to confirming these two conjectures for sufficiently large n. Recently, Fang et al. (2024) [12] characterized the extremal graph with ${\text{spex}}_P(n,t{C}_l)$ in planar graphs by using this key. In this paper, we first focus on outerplanar graphs and adopt a similar approach to describe the key structure of the connected extremal graph with ${\text{spex}}_{\mathrm{OP}}(n,F)$, where F is contained in $K_1\vee P_{n-1}$ but not in $K_1\vee \left(\right(t-1)K_2\cup (n-2t+1\left)K_1\right)$. Based on this structure, we determine ${\text{spex}}_{\mathrm{OP}}(n,B_{t,l})$ and ${\text{spex}}_{\mathrm{OP}}(n,(t+1\left)K_2\right)$ along with their unique extremal graphs for all $t\ge 1$, $l\ge 3$ and large n. Moreover, we further extend the results to planar graphs, characterizing the unique extremal graph with ${\text{spex}}_P(n,B_{t,l})$ for all $t\ge 3$, $l\ge 3$ and large n.