Characterization of regular bipolar fuzzy graphs

In this paper, adjacency sequence of a vertex, first and second fundamental sequences are defined in a bipolar fuzzy graph with example. Some examples are constructed to show that if G is a regular bipolar fuzzy graph (BFG), the underlying crisp graph need not be regular and all the vertices need not have the same adjacency sequence. Also it is shown that if G and its underlying crisp graph are regular, all the vertices need not have the same adjacency sequence. A necessary and sufficient condition is established for a BFG with at most four vertices to be regular using the concept of adjacency sequences. Moreover, some characterizations have been made for a line graph of a regular BFG to be regular, the complement of a regular BFG to be regular, etc.

Automated summary

This paper defines the adjacency sequence of a vertex, as well as first and second fundamental sequences in a bipolar fuzzy graph, and presents examples to illustrate the relationships between regular BFG and its underlying crisp graph. It establishes necessary and sufficient conditions for a BFG with at most four vertices to be regular using the concept of adjacency sequences. Additionally, characterizations are made for a line graph of a regular BFG to be regular, the complement of a regular BFG to be regular, etc.