The maximum spectral radius of outerplanar and planar $k$-uniform hypergraphs

Abstract

For $k\ge3$, a $k$-angulation is a simple $2$-connected outerplane graph whose interior faces are bounded by $k$-cycles, and a closed $k$-angulation is a simple $2$-connected plane graph all of whose faces are bounded by $k$-cycles. Their face hypergraphs have the same vertices as the graphs; the hyperedges are the vertex sets of the interior faces, respectively all faces. For $k=3$, the extremal problems over these hypergraphs agree with those for the outerplanar and planar $3$-uniform hypergraphs of Ellingham, Lu and Wang, who determined the outerplanar maximizer for large $n$ and conjectured the planar one. In this paper, for each fixed $k$, we determine the hypergraphs of maximum spectral radius in both classes for all sufficiently large $n$ satisfying the corresponding face-count divisibility condition. In the outerplanar case, the unique extremal hypergraph is the face hypergraph of the fan, in which one vertex lies on every interior face. In the planar case with $k=3$, it is the face hypergraph of $K_2+P_{n-2}$, which confirms the conjecture of Ellingham, Lu and Wang. For $k\ge4$, the extremal hypergraphs are the face hypergraphs of balanced theta graphs, in which two vertices lie on every face. For $k=4$, the result holds for every $n\ge5$, with extremal hypergraph ${\mathcal H}(K_{2,n-2})$. For $k\ge6$ and an even number of faces, there are exactly $\lfloor(k-2)/2\rfloor$ extremal hypergraphs up to isomorphism.