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Research & Reviews: Discrete Mathematical Structures Original Research

Intuitionistic Fuzzy Hypergraph Laplacians and Dominating Transversals for Resilient Discrete Network Design

  1. Markala Karthik 1Department of Electrical and Electronics Engineering, SR University, Warangal
  2. S. Vairachilai School of Computer Science and Artificial Intelligence, SR University, Warangal
  3. Mohammed Almakki School of Engineering, Architecture and Interior Design, Amity University Dubai
  4. Mohammed El Khider Department of General Undergraduate Curriculum Requirements, University of Dubai

Abstract

This manuscript develops a discrete mathematical framework for resilience analysis on networks whose interactions are polyadic, uncertain, and partially conflicting. Classical graphs compress multi-way coordination into pairwise edges, while ordinary fuzzy graphs often ignore the non-membership information that becomes critical in emergency logistics, infrastructure interdependence, and cyberphysical coordination. We therefore formulate an intuitionistic fuzzy hypergraph in which each vertex hyperedge incidence carries membership, non-membership, and hesitation, and we construct a hesitation-aware Laplacian that remains symmetric and positive semidefinite. On this basis, two coupled optimization objects are introduced: an uncertainty-regularized algebraic connectivity that quantifies structural robustness, and a dominating transversal number that quantifies the minimum control set needed to supervise all higher-order interactions. We derive lower and upper bounds linking the second Laplacian eigenvalue, weighted vertex degrees, and domination cost; establish a perturbation estimate showing how hesitation inflation degrades connectivity; and propose a greedy spectral-transversal algorithm with near-linear practical complexity in sparse hypergraphs. A synthetic benchmark on modular response networks shows that the proposed operator separates fragile and resilient designs more sharply than pairwise surrogates and identifies low-cardinality control sets with lower uncertainty burden. The paper is aligned with discrete structures, logic-oriented uncertainty, set-theoretic modeling, relations, functions, and matrix-based analysis, while also incorporating recent intuitionistic fuzzy, hypergraph, and fuzzy optimization literature. The resulting framework is intended as a submission-ready theoretical and computational study for a journal audience in discrete mathematical structures.

Keywords

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