A Spectral Hilton--Milner--Frankl Theorem for $t$-Intersecting Families

Abstract

Keevash, Lenz, and Mubayi proved a spectral Erd\H{o}s--Ko--Rado theorem, showing that, for sufficiently large $n$, the complete $t$-star uniquely maximizes the adjacency-tensor spectral radius among all $t$-intersecting $k$-uniform families. In this paper, we establish a spectral Hilton--Milner--Frankl theorem for nontrivial $t$-intersecting families in the explicit range $1\le t\le k-2$ and $n\ge 100\cdot 2^k k^7$. More precisely, we prove that, for every nontrivial $t$-intersecting $k$-uniform family $\mathcal F$, the spectral radius satisfies \[ \rho(\mathcal F)\le \max\{\rho(\mathcal H_{n,k,t}),\rho(\mathcal A_{n,k,t})\}, \] where $\mathcal H_{n,k,t}$ and $\mathcal A_{n,k,t}$ are the two extremal families appearing in the classical Hilton--Milner--Frankl theorem. Moreover, equality holds only for the extremal candidates attaining the maximum, up to isomorphism. We further compare the two candidates asymptotically. For each fixed $t$, the unique real solution $x=x_t$ of \[ (t+2)^{x-t-1}(t+1)^{t+1}=(x-t+1)^{x-1} \] determines, as $k$ varies, which of $\mathcal H_{n,k,t}$ and $\mathcal A_{n,k,t}$ has the larger asymptotic spectral radius.