Transformation of Energy in Fuzzy Graph and Its Application in Ecological System

This paper discusses the computation of graph energy of a fuzzy graph when edge weights are represented by interval numbers. The adjacency matrix is first constructed for the given fuzzy graph, where each edge weight is an interval number. The center values of these intervals are then calculated, forming a new matrix whose elements are these center values. The eigenvalues of this matrix are determined, and from them, the energy of the graph is computed. In the second part of the paper, an application to ecological systems is discussed. It demonstrates that any ecological system, particularly a food web, can be modeled as a fuzzy graph with interval-weighted edges. Since species populations and energy production levels are inherently uncertain, these values are represented as interval numbers. The food web is structured from its lower levels (consisting of plants and chemical nutrients) to its upper levels, comprising various consumers. Finally, the total energy within the ecological system is obtained by summing the energy across all levels. This is the first attempt to model an ecological system with the help of a fuzzy graph and try to find out the transformation of energy from level to level. To our knowledge, no work has been available to model the ecological system.