The Hypertree with the Largest Spectral Radius Among all Hypertrees with Given Size and Stability Number

Abstract

In this paper, we determine the unique uniform hypertree with the maximum spectral radius in the set (denoted by $${\cal T}(m,r,\alpha)$$ ) of all r-uniform hypertrees with given size m(> r) and stability number α. More precisely, for any $$T \in {\cal T}$$ with m > r, we have $$\rho (T) \le {\left({{1 \over {1 - {\alpha _0}}}} \right)^{1/r}},$$ where α0 is the maximum root of $${x^{r - 1}}\left({{1 \over {1 - x}} - {1 \over {{x^s}}} - l} \right) = g,$$ and integers g, l, s are determined by n − 1 − α = g(r − 1) + s, where n = m(r − 1) + 1 is the order of T and 0 ≤ s < r − 1, and l = m − gr − s − 1. Further, $$\rho (T) = {({1 \over {1 - {\alpha _0}}})^{1/r}}$$ if and only if T = R((r − 1)g; 0l; s).