On the Domination Energy of k -Uniform Hypergraphs with Applications to Supply Chain Resilience

Abstract

The concept of graph energy, defined as the sum of the absolute eigenvalues of a graph's adjacency matrix, has been widely studied for its applications in chemistry and network theory. In this paper, we extend this notion to k-uniform hypergraphs by introducing the domination energy, a spectral invariant derived from a hypergraph's minimum dominating set. We introduce the domination matrix of a hypergraph, establish theoretical bounds for its energy, and explore its combinatorial properties. Furthermore, we demonstrate practical applications of this framework in supply chain risk management. By modeling multi-company production processes as hyperedges in a multi-layer hypergraph, we develop a mathematical framework for identifying critical companies whose disruption could paralyze entire supply chains. We develop algorithms with provable approximation guarantees, quantitative criticality metrics, and a tiered mitigation framework. This work bridges spectral hypergraph theory with real world complex system analysis, offering both theoretical contributions and practical tools for enhancing supply chain resilience. Since the domination matrix is a symmetric shift operator acting on signals supported on the hypergraph, the domination energy belongs to the family of spectral descriptors employed in graph and hypergraph signal processing and in multiscale network analysis. Consequently, the bounds established in this paper serve as structural information measures for higher-order networks and as groundwork for multiresolution methods on hypergraphs.