Spectral extremal problems on outerplanar and planar graphs
Xilong Yin, Dan Li
TL;DR
We address Brualdi-Solheid style questions for outerplanar and planar graphs by determining the maximum spectral radius among $F$-free graphs, denoted $\text{spex}_{\mathcal{OP}}(n,F)$ and $\text{spex}_{\mathcal{P}}(n,F)$. A central structural result shows that, for $F$ contained in $K_1 \vee P_{n-1}$ but not in $K_1 \vee ((t-1)K_2\cup(n-2t+1)K_1)$, any connected extremal outerplanar graph $G$ has a dominating vertex $u$ with $N_G(u)=V(G)\setminus\{u\}$ and $x_u=1$, yielding $G\cong K_1 \vee H$ with $H$ a disjoint union of paths; this reduces the problem to analyzing joins with path forests. Using this framework, the paper exactly identifies the extremal graphs and proves uniqueness for $\text{spex}_{\mathcal{OP}}(n,B_{tl})$ and $\text{spex}_{\mathcal{OP}}(n,(t+1)K_{2})$ for all $t\ge1$, $l\ge3$ and sufficiently large $n$, and extends the results to planar graphs by determining the unique extremal for $\text{spex}_{\mathcal{P}}(n,B_{tl})$ when $t\ge3$, $l\ge3$ and large $n$. The proofs combine Perron–Frobenius spectral theory, Rayleigh quotients, path-decomposition arguments, and $(s_1,s_2)$-transformations to obtain explicit extremal constructions such as $K_1 \vee H_{\mathcal{OP}}(\cdot,\cdot)$ and $K_2 \vee H_{\mathcal{P}}(\cdot,\cdot)$, extending prior results on $tC_\ell$ and related forbidden structures in outerplanar and planar graphs. Overall, the results advance the understanding of spectral extremal problems in graph classes by delivering complete descriptions of extremal graphs for large $n$ and key forbidden subgraphs.
Abstract
Let $\emph{spex}_{\mathcal{OP}}(n,F)$ and $\emph{spex}_{\mathcal{P}}(n,F)$ be the maximum spectral radius over all $n$-vertex $F$-free outerplanar graphs and planar graphs, respectively. Define $tC_l$ as $t$ vertex-disjoint $l$-cycles, $B_{tl}$ as the graph obtained by sharing a common vertex among $t$ edge-disjoint $l$-cycles %$B_{tl}$ as the graph obtained by connecting all cycles in $tC_l$ at a single vertex, 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 [J. Combin. Theory Ser. B, 2017] determined the fundamental structure as the key to confirming these two conjectures for sufficiently large $n.$ Recently, Fang et al. [J. Graph Theory, 2024] characterized the extremal graph with $\emph{spex}_{\mathcal{P}}(n,tC_l)$ 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 $\emph{spex}_{\mathcal{OP}}(n,F)$, where $F$ is contained in $K_1 \vee P_{n-1}$ but not in $K_{1} \vee ((t-1)K_2\cup(n-2t+1)K_1)$. Based on this structure, we determine $\emph{spex}_{\mathcal{OP}}(n,B_{tl})$ and $\emph{spex}_{\mathcal{OP}}(n,(t+1)K_{2})$ along with their unique extremal graphs for all $t\geq1$, $l\geq3$ and large $n$. Moreover, we further extend the results to planar graphs, characterizing the unique extremal graph with $\emph{spex}_{\mathcal{P}}(n,B_{tl})$ for all $t\geq3$, $l\geq3$ and large $n$.