Spectral extremal problems on outerplanar and planar graphs

TL;DR

This paper determines the maximum spectral radius for large $n$-vertex outerplanar and planar graphs avoiding certain subgraphs, identifying unique extremal graphs and extending previous conjectures and results in spectral graph theory.

Contribution

It characterizes the extremal graphs with maximum spectral radius in outerplanar and planar graphs for various forbidden subgraphs, extending known results and confirming conjectures for large $n$.

Findings

Determined spectral extremal values for outerplanar graphs avoiding specific subgraphs.

Identified unique extremal graphs for planar graphs with certain forbidden subgraphs.

Extended previous conjectures and characterized extremal structures for large graphs.

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, Cvetkovi\'c 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 $n$.