Planar and Outerplanar Spectral Extremal Problems based on Paths
Xilong Yin, Dan Li, Jixiang Meng
TL;DR
The paper addresses the problem of determining extremal graphs that maximize the spectral radius $\rho(G)$ among $n$-vertex $F$-free planar or outerplanar graphs for forbidden subgraphs related to paths, specifically $F \in \{P_{t\cdot l}, tP_l\}$. It extends structural frameworks from Tait–Tobin and recent path-based planar results to identify exact extremal graphs for sufficiently large $n$, showing that outerplanar extremals take the form $G \cong K_1 \vee H_{\mathcal{OP}}(\lceil (l-2)/2\rceil,\lfloor (l-2)/2\rfloor)$ (for appropriate cases) while planar extremals take the form $G \cong K_2 \vee H_{\mathcal{P}}(tl-l-1,l-1)$, with several variants depending on $t$ and $l$. The authors develop and apply path-transformations and eigenvalue tools to prove these results, and also derive corollaries for special cases such as $l=2$, as well as edge-disjoint path variants. The work advances spectral Turán theory under planarity constraints and provides concrete structural templates that may guide future generalizations to broader families of forbidden subgraphs. It also surfaces open problems for unresolved cases (notably $t=3$) and proposes conjectures for the precise extremal graphs in those regimes.
Abstract
Let SPEX$_\mathcal{P}(n,F)$ and SPEX$_\mathcal{OP}(n,F)$ denote the sets of graphs with the maximum spectral radius over all $n$-vertex $F$-free planar and outerplanar graphs, respectively. Define $tP_l$ as a linear forest of $t$ vertex-disjoint $l$-paths and $P_{t\cdot l}$ as a starlike tree with $t$ branches of length $l-1$. Building on the structural framework by Tait and Tobin [J. Combin. Theory Ser. B, 2017] and the works of Fang, Lin and Shi [J. Graph Theory, 2024] on the planar spectral extremal graphs without vertex-disjoint cycles, this paper determines the extremal graphs in $\text{SPEX}_\mathcal{P}(n,tP_l)$ and $\text{SPEX}_\mathcal{OP}(n,tP_l)$ for sufficiently large $n$. When $t=1$, since $tP_l$ is a path of a specific length, our results adapt Nikiforov's findings [Linear Algebra Appl. 2010] under the (outer)planarity condition. When $l=2$, note that $tP_l$ consists of $t$ independent $K_2$, then as a corollary, we generalize the results of Wang, Huang and Lin [arXiv: 2402.16419] and Yin and Li [arXiv:2409.18598v2]. Moreover, motivated by results of Zhai and Liu [Adv. in Appl. Math, 2024], we consider the extremal problems for edge-disjoint paths and determine the extremal graphs in $\text{SPEX}_\mathcal{P}(n,P_{t\cdot l})$ and $\text{SPEX}_\mathcal{OP}(n,P_{t\cdot l})$ for sufficiently large $n$.