Journal
AFRIKA MATEMATIKA
Volume 32, Issue 5-6, Pages 1043-1057Publisher
SPRINGER HEIDELBERG
DOI: 10.1007/s13370-021-00880-y
Keywords
Bipolar fuzzy graph; Adjacency sequence; Fundamental sequence; Regular BFG; Line graph; Complement
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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.
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.
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