Maximum spectral radius of outerplanar 3‐uniform hypergraphs

In this paper, we study the maximum spectral radius of outerplanar 3‐uniform hypergraphs. Given a hypergraphℋ ${\rm{ {\mathcal H} }}$ , the shadow ofℋ ${\rm{ {\mathcal H} }}$ is a graphG $G$ withV ( G ) = V ( ℋ )$V(G)=V({\rm{ {\mathcal H} }})$ andE ( G ) = { u v : u v ∈ h for some h ∈ E ( H ) }$E(G)=\{uv:uv\in \,h\,\text{for\unicode{x02007}some}\,h\in E({\mathscr{H}})\}$ . A graph is outerplanar if it can be embedded in the plane such that all its vertices lie on the outer face. A 3‐uniform hypergraphℋ ${\rm{ {\mathcal H} }}$ is called outerplanar if its shadow has an outerplanar embedding such that every hyperedge ofℋ ${\rm{ {\mathcal H} }}$ is the vertex set of an interior triangular face of the shadow. Cvetković and Rowlinson conjectured in 1990 that among all outerplanar graphs onn $n$ vertices, the graphK 1 + P n − 1${K}_{1}+{P}_{n-1}$ attains the maximum spectral radius. We show a hypergraph analogue of the Cvetković–Rowlinson conjecture. In particular, we show that for sufficiently largen $n$ , then $n$ ‐vertex outerplanar 3‐uniform hypergraph of maximum spectral radius is the unique 3‐uniform hypergraph whose shadow isK 1 + P n − 1${K}_{1}+{P}_{n-1}$ .