Distance Degree Sequence and Distance Neighborhood Degree Sequence of Vertices in Fuzzy Graphs and Some of Their Properties

Objectives: The objective of this study is to explore new types of sequences in fuzzy graphs namely distance degree sequence and distance neighborhood degree sequence of fuzzy graphs which may have various applications in the real world. Methods: In this paper, two unique sequences for each vertex of fuzzy graphs are given by means of its degree and neighborhood degree which are called as the distance degree sequence and distance neighborhood degree sequence of vertices of fuzzy graphs respectively. The n-tuple of the distance degree sequence of vertices of a fuzzy graph arranged in lexicographic order is called the distance degree sequence of the fuzzy graph and the n-tuple of distance neighborhood degree sequence of vertices of a fuzzy graph arranged in lexicographic order is called the distance neighborhood degree sequence of the fuzzy graph. Findings: The distance degree sequence and distance neighborhood degree sequence of fuzzy graphs are introduced. The properties of these sequences and the general forms of these sequences of various fuzzy graphs are given. The relationship between the distance degree sequence and distance neighborhood degree sequences is discussed. Novelty: The distance degree sequence of graphs has already been introduced. In fuzzy graph theory, the distance degree sequence and distance neighborhood degree sequence are introduced in this paper. Mathematics Subject Classification (202 0): 03E72, 05C72, 05C12. Keywords: Fuzzy graphs, Distance degree sequence, Distance neighborhood-degree sequence, Regular fuzzy graphs