The maximum spectral radius of outerplanar 3-uniform hypergraphs with a given size

Abstract

Given a 3-uniform hypergraph $\mathcal{G}=\big(V(\mathcal{G}),E(\mathcal{G})\big)$, the shadow of $\mathcal{G}$ is a graph $G=\big(V(G),E(G)\big)$ with $V(G) = V(\mathcal{G})$ and $E(G) = \{uv: uv \in e \text{ for some $e$ }\in E(\mathcal{G})\}$.
A graph is planar if it can be drawn on the plane such that its edges intersect only at their endpoints.
A planar graph is outerplanar if it has a planar embedding such that all its vertices lie on outer face.
If the shadow of $\mathcal{G}$ is an (outer)planar graph, then $\mathcal{G}$ is a $3$-uniform (outer)planar hypergraph.
Let $P_1+P_{n-1}$ be the join of $P_{1}$ and $P_{n-1}$, where $P_n$ is a path with $n$ vertices and $n\geqslant 3$.
Ellingham et al.\ (2022) proved that for sufficiently large $n$,
the $n$-vertex outerplanar 3-uniform hypergraph of maximum spectral radius is the unique 3-uniform hypergraph whose shadow is $P_1+P_{n-1}$.
In this paper, we prove that for sufficiently large $m$, the $m$-edge outerplanar
3-uniform hypergraph of maximum spectral radius is also the unique 3-uniform hypergraph whose shadow is $P_1+P_{m+1}$.