The Complex Interval-Valued q-Rung Orthopair Fuzzy Set (CIVq-RungFS), is a generalization of Complex Interval-Valued Intuitionistic Fuzzy Set (CIVIFS) and Complex Interval-Valued Pythagorean Fuzzy Set. It provides a flexible model for expressing uncertainty and vagueness through the interval-valued truth and falsity grades. It is composed of both complex interval-valued membership and complex interval-valued non-membership degrees, adhering to the axioms that \(0\le (\nu _{M_{1}}^+(p))^q+(\nu _{M_{2}}^+(p))^q\le 1,~0\le (\chi _{M_{1}}^+(p))^q+(\chi _{M_{1}}^+(p))^q\le 2\pi ,~ q\ge 1.\) The complex interval-valued q-rung orthopair fuzzy model is used to express complex two-dimensional information. The future applicability of the suggested model is essentially guaranteed. This research integrates graph structures with CIVq-RungFS, by introducing the concept of Complex Interval-Valued q-Rung Orthopair Fuzzy Graph Structure (CIVq-RungFGS). Additionally, it proposes an innovative idea for the dominating energy of graphs, inspired by the newly presented CIVq-RungFGS concept. More precisely, with the aid of illustrative examples, the adjacency matrix of a dominating CIVq-RungFGS, the spectrum of the adjacency matrix, and their associated theory are constructed. Moreover, investigation is done on some features and constraints for the energy of dominating and \(\mu _J\) -Inverse dominating signless Laplacian Energy graph structures in the CIVq-RungFS environment. Moreover, the concept of isomorphic properties related to energy and the inverse dominance of signless Laplacian energy in CIVq-RungFGS are studied. Finally, it explores the potential applications of CIVq-RungFS with graph structure methodology in healthcare facility design. A formula has also been created to make our application’s basic operations more understandable.
