Energy of interval-valued fuzzy graphs and its application in ecological systems

Abstract

Many papers have been published for finding eigenvalues and eigenvectors of a fuzzy matrix (FM) (i.e. matrix with membership values) as well a fuzzy graph. But, all papers determine such parameters based on conventional arithmetic operators, though the valid operations on FMs are max–min. To the best of our knowledge no papers are published to find eigenvalues and eigenvectors based on max–min operators. In this paper, a novel technique is adapted to find the eigenvalues and eigenvectors of an interval-valued fuzzy graph (IVFG) using max–min operators. The energy of an IVFG is defined and computed using max–min operators. Finally, an application of eigenvalues of an IVFG is discussed for the ecological system. In ecology, the amount of food consumed by a predator from the preys is represented as an interval valued fuzzy membership values which is natural as the consumption of food for a predator from preys is uncertain. So this application is very much appropriate for eigenvalues and eigenvectors as well as energy of an IVFG.