The maximum spectral radii of weighted uniform loose cycles and unicyclic hypergraphs

Abstract

  A weighted $k$-uniform loose cycle of length $m$, denoted by $C_{m,k}$, is a cyclic list of weighted edges $e_1, e_2,\ldots, e_m$ such that consecutive edges intersect in exactly one vertex, and  nonconsecutive edges are disjoint, where $|e_i|=k$ for all $1 \leq i\leq m$. For a given positive weight set, we determine the distribution of weights of $C_{m,k}$ with the maximum spectral radius. Moreover, we characterize the unique weighted hypergraph with the maximum spectral radius in the class of all weighted uniform unicyclic hypergraphs with a given positive weight set.