The maximum spectral radius of uniform hypergraphs whose shadow excludes a complete or complete bipartite minor

Abstract

For a $k$-uniform hypergraph $\mathcal{H}$, the shadow of $\mathcal{H}$ is the graph whose edges are the pairs covered by a hyperedge. In this paper, for all sufficiently large $n$, we determine the $n$-vertex $k$-uniform hypergraphs of maximum adjacency-tensor spectral radius whose shadow has no $K_t$ minor, for every $t\ge k+1$, and those whose shadow has no $K_{s,t}$ minor, for every $2\le s\le t$ with $s+t\ge k+1$ and every residue of $n-s+1$ modulo $t$; outside these ranges the problems are trivial. This extends to uniform hypergraphs the theorem of Tait on graphs with no $K_r$ or $K_{s,t}$ minor, whose remaining residues were settled by Zhai and Lin. In each case the extremal hypergraph is unique, and it is the $k$-clique hypergraph of the join of a clique with a graph that we call the light part. For $K_{s,t}$ the answer depends on $j=k-s+1$. When $j\le 1$, the maximum has order $n^{(k-1)/k}$, and the light part is the one found by Zhai and Lin for the adjacency matrix, including its exceptional components. When $j\ge 2$, a regime that does not occur for graphs, the maximum has order $n^{(s-1)/k}$ and $t$ enters its leading constant. The light part then consists of copies of $K_t$ and one smaller clique, with a single exception: for $(k,s,t)=(9,8,8)$ and $n-s+1\equiv 2(\mathrm{mod}8)$, the complement of the Petersen graph appears. When the smaller clique has between $1$ and $j-1$ vertices, the extremal graph is not unique. In particular, for $t=8$, $4\le s\le 7$, $k=s+1$ and $n-s+1\equiv 2(\mathrm{mod}8)$, the clique hypergraph of the extremal graph of Zhai and Lin is not extremal. For $j\ge 2$ the light part is determined by a weighted clique inequality, which for $j\ge 3$ follows from a weighted form of the closed-neighborhood counting of Chao and Dong.

50 pages