Extremal spectral result of outerplanar graphs without $P_{3\cdot l}$

Abstract

A graph $G$ is $F$-free if it does not contain $F$ as a subgraph. Let $\mathrm{spex}(n,F)$ be the maximum spectral radius over all $n$-vertex $F$-free outerplanar graphs. For integers $t\ge 1$ and $l\ge 2$, let $P_{t\cdot l}$ be the starlike tree with $t$ branches of length $l-1$. For sufficiently large $n$, Yin, Li, and Meng [arXiv:2504.04364v1] characterized the unique extremal graph for $\mathrm{spex}(n,P_{t\cdot l})$ when $t=1$, $t=2$, or $t\ge 4$. They left the case $t=3$ open and proposed a natural candidate for the extremal graph. We show that this candidate is not extremal and determine the unique extremal graph for $\mathrm{spex}(n,P_{3\cdot l})$. For every $l\ge 3$ and all sufficiently large $n$, this unique extremal graph is $K_1\vee (2P_{2l-3}\cup q{P}_{l-2}\cup P_r),$ where $q$ and $r$ are integers satisfying $n=2(2l-3)+q(l-2)+r+1,$ $q\ge 0,$ $0\le r<l-2.$

10 pages