Aug 2026· International Journal of Wavelets, Multiresolution and Information Processing· 0 citations
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.
This study explores the spectral characteristics and energy distributions associated with selected graph operations derived from the first Zagreb, second Zagreb, and sum-connectivity matrices to contribute to understanding how algebraic operations induce spectral energy shifts analogous to perturbations in physical or molecular graph systems.
S. Sripriya, A. Anuradha· Baghdad Science Journal· 0 citations
We study per-colour independent (k)-rainbow domination and its connection with independent domination in generalized prisms. Building on the known prism identity and the trivial regime above the maximum degree, we focus on the boundary case where the number of colours equals the maximum degree. For every fixed (k\ge 3), we prove that the decision problem remains NP-complete even on a highly restricted class of graphs: (C_4)-free, bipartite, ((k,2))-biregular subdivision graphs arising from simple (k)-regular graphs. The reduction gives an exact correspondence between optimal rainbow-independent dominating functions on the subdivision graph and proper (k)-edge-colourings of the original graph. We also introduce an excess parameter measuring how far the domination number lies above its natural lower bound. For cubic graphs, this excess coincides with the classical edge-colouring degree and therefore with standard resistance parameters for subcubic graphs. These results reveal a sharp one-unit threshold: above the maximum degree the problem becomes trivial for every graph, while at the boundary NP-hard instances already occur within a very narrow structural family.
The ‘divide and conquer’ paradigm proves to be one of the most frequently used techniques for dealing with the complexities of graph-related problems. Therefore, it is of great importance to measure the tendency of a vertex to be critical and its susceptibility in a graph. The criticality of a vertex is often analysed in terms of its strength. Removing a highly critical vertex from a graph modelling a network may introduce vulnerability into the system represented by the graph. Minimizing the vulnerability of such a network without affecting its fundamental structure, thereby improving the stability of the graph, is the primary objective of the article. To achieve this, certain properties of Euler graphs are analysed in terms of vertex strength, and a method is presented for determining all possible constructions of Euler graphs corresponding to different integer partitions. The parts of a partition represent the vertex strengths, and their sum corresponds to the total vertex strength of the graph. Various connectivity indices are employed to validate the proposed constructions. Furthermore, their interrelationships and potential real-life applications are also discussed. It is evident from the constructions that they may play a vital role in developing network deception technology to protect digital assets, as each partition of the network generates a distinct network.
Saifur Rahman, Raju Doley· Acta Universitatis Sapientia...· 0 citations
Bipolar fuzzy graphs (BFGs) extend classical fuzzy graph theory by in-corporating both positive and negative membership degrees, enabling the representation of dual-aspect uncertainty in complex systems. Product bipo-lar fuzzy graphs (PBfGs) provide a refined framework for modeling interde-pendent relationships where edge strength is determined muftipficatirefy by vertex attributes, rather than by the classical minimum or maximum opera-tors. This paper formulates and investigates four key advanced domination variants within the strict product-based constraints of PBfGs: secure dom-ination, which ensures network resilience under node failure; 2-domination, which guarantees redundant coverage for fault tolerance; connected perfect domination, which enforces structural connectivity and exactness of cover-age; and tiiùofe edge domination, which extends control mechanisms to edge-based network architectures. We establish theoretical foundations, prove characterization theorems and bounds, examine operational properties un-der graph products (Cartesian product, composition, union, and join), and demonstrate a real-world application to secure transit network design.
Mujeeburahman T. C., R. Theivaraman, K. Maheshwaran et al.· Adolescência e Saúde· 0 citations
This paper develops an exact algorithm based on the Branch and Bound approach for solving PIDS on chordal graphs, which involves identifying the smallest group of vertices in a given network that maximizes influence throughout the network.
Y A Bekhti, M. Lalou, Méziane Aïder et al.· Pesquisa Operacional· 0 citations
In this paper, we initiate the study of colored exponential random graph models (ERGMs), a class of exponential-family models for networks with multiple types of edge relations. Using the framework of probability graphons, we first derive a variational representation for the limiting free energy, whose maximizers determine the asymptotic structure of typical samples from the model. Then we identify several general families of colored ERGMs exhibiting replica symmetry, where the variational problem has constant maximizers and the model asymptotically concentrates on product colorings with independent edges. For general colored ERGMs, we derive Euler-Lagrange fixed-point equations for the variational maximizers, which in turn yield a general high-temperature uniqueness criterion. In the complementary zero-temperature regime, we establish a two-level selection principle: the leading energy term determines the ground states, while the lower-order energy terms, combined with entropy, act as a tie-breaker to determine the asymptotic zero-temperature structure of the model. We illustrate this principle through the induced wedge and rainbow triangle ERGMs. Both models have natural interpretations in multitype networks, and their zero-temperature limits exhibit interesting structures that connect to well-known results in extremal combinatorics. We further establish finite-temperature symmetry breaking for both these models and complement the rigorous results with numerical experiments.
B. Bhattacharya, Pierfrancesco Dionigi, Ankana Ganguly et al.· 0 citations
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