Eigen Spectrum of k− Uniform Loose Cyclic Hypergraphs

Abstract

Hypergraphs extend ordinary graphs by allowing a hyperedge to connect more than two vertices. A hypergraph is k -uniform when each hyperedge contains exactly k vertices, and it is loose cyclic when the hyperedges are arranged cyclically so that consecutive hyperedges share exactly one vertex while non-consecutive hyperedges are disjoint. This study examines the possible k-uniform loose cyclic hypergraphs in relation to the number of vertices and develops a computational procedure for determining their spectral properties. For a loose cyclic hypergraph H = (V, E)  with n vertices and m hyperedges, the relation n = m (k-1) is used to describe admissible configurations. An adjacency matrix is formed by assigning each off-diagonal entry according to the number of hyperedges containing the corresponding pair of vertices. A Python-based procedure is then used to construct the adjacency matrix for admissible parameter choices and to compute its eigenvalues and eigenvectors. The method is illustrated using a 4-uniform loose cyclic hypergraph on 15 vertices with five hyperedges. The resulting 15 × 15 adjacency matrix and its eigenvalues demonstrate the computational implementation of the procedure. The study provides a systematic matrix-based approach for obtaining the eigen spectrum of uniform loose cyclic hypergraphs when closed-form expressions are difficult to derive, while retaining the structural conditions that define the loose cyclic arrangement.