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

A weighted k-uniform loose cycle of length m, denoted by C-m,C-k, is a cyclic list of weighted edges e(1), e(2), ... , e(m) such that consecutive edges intersect in exactly one vertex, and nonconsecutive edges are disjoint, where |e(i)| = k for all 1 <= i <= m. For a given positive weight set, we determine the distribution of weights of C-m,C-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.