<p>Picture fuzzy graphs effectively represent complex problems that cannot be accurately modeled using traditional fuzzy graphs and intuitionistic fuzzy graphs. The incorporation of neutrality degree significantly enhances the representation of uncertain membership values. In the context of energy applications, dominating energy plays a crucial role. Therefore, study of dominating energy in picture fuzzy graphs give more accurate results than other existing methods in the field of energy. In this article, we introduced the notion of dominating energy in picture fuzzy graphs. Picture fuzzy dominating adjacency matrix, eigen values, spectrum of picture fuzzy graphs are presented with examples. Dominating set and domination number of picture fuzzy graph is introduced. Dominating degree of a vertex, weight of a vertex is presented. Energy arithmetic mean is introduced. Apart from these, dominating energy in various operations like, union, join, complement of picture fuzzy graphs are developed. Also, some theorems regarding dominating energy are derived in this paper. We calculated lower and upper bound of dominating energy. Finally, a practical application of dominating energy is demonstrated through fund distribution among various sports. This innovative approach enables more effective resource allocation and decision-making in complex scenarios.</p>

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Picture Fuzzy Graphs with Dominating Energy: A New Perspective for Solving Complex Problem

  • Avisek Banerjee,
  • Sk Amanathulla

摘要

Picture fuzzy graphs effectively represent complex problems that cannot be accurately modeled using traditional fuzzy graphs and intuitionistic fuzzy graphs. The incorporation of neutrality degree significantly enhances the representation of uncertain membership values. In the context of energy applications, dominating energy plays a crucial role. Therefore, study of dominating energy in picture fuzzy graphs give more accurate results than other existing methods in the field of energy. In this article, we introduced the notion of dominating energy in picture fuzzy graphs. Picture fuzzy dominating adjacency matrix, eigen values, spectrum of picture fuzzy graphs are presented with examples. Dominating set and domination number of picture fuzzy graph is introduced. Dominating degree of a vertex, weight of a vertex is presented. Energy arithmetic mean is introduced. Apart from these, dominating energy in various operations like, union, join, complement of picture fuzzy graphs are developed. Also, some theorems regarding dominating energy are derived in this paper. We calculated lower and upper bound of dominating energy. Finally, a practical application of dominating energy is demonstrated through fund distribution among various sports. This innovative approach enables more effective resource allocation and decision-making in complex scenarios.