Characterizations of the Direct Sum of Two Difference - Mean Fuzzy Graphs

Objectives: This study presents a new type of fuzzy graph known as the difference mean fuzzy graph by introducing difference mean edge. Methodology: In this paper, difference mean edge in a fuzzy graph is defined by considering the relationship between the membership value of the edge and the membership values of its end vertices. Also, difference mean fuzzy graph is defined and its properties are derived. Findings: The difference mean edge and the difference mean fuzzy graph are introduced. The requirements for an edge in the direct sum of two fuzzy graphs to be a difference mean edge are found in this study. Additionally, conditions are derived such that the direct sum of two fuzzy graphs is a difference mean fuzzy graph. Novelty: Depending on the membership values of the edges and vertices, effective edge in fuzzy graph have already been defined. A new concept of difference mean edge in fuzzy graph is introduced. Using this, difference mean fuzzy graph is also introduced. Characterizations of the difference mean edge in the direct sum of fuzzy graphs are attained. The requirements for the necessary and sufficient component of difference mean fuzzy graphs to be a direct sum are suggested. Mathematics Subject Classification (2020): 05C72, 05C76. Keywords: Difference mean edge, Difference Mean fuzzy graph, Effective fuzzy graph, Effective difference mean edge, Direct sum