Results on a Chromatic Number of a Bi-polar Fuzzy Complete Bipartite Graphs and Labelling of Tri-Polar Fuzzy Graphs

The ultimate objective of a piece of research work is to present the labelling of vertices in 3-PFG and labelling of distances in 3-PFG. Also, we characterize some of its properties. Later, we define the vertex and edge chromatic number BFComplete Bipartite graph. Further we illustrated an example for BFRGS which represents a Route Network system.


Introduction
The conventional investigation of SG starts in the mid-20 th century. SGs are fundamental logarithmic models in numerous parts of designing formal dialects, in coding, Finite State Machine, automaton. The significant part of graph theory in computer presentations is the improvement of calculations in graphs. A graph structure is a helpful tool in cracking the combinatory problems in diverse areas of computer science containing clustering, image capturing, data mining, image segmentation, networking, and computational intelligence systems. (Zadeh. L. A.,1965) presented the fuzzy theory. Further, (Rosenfeld., 1971) applied it to the classical theory of subgroups. Later, (Bhargavi. Y., 2020) and (Mordeson J. N2003) developed the classical theory of fuzzy graphs, SGs. Accepting the above examination as starting point, in this Research article present the labelling of vertices in 3-PFGand labelling of distances in 3-PFG. Also, we characterize some of its properties. Later, we define the vertex and edge chromatic number BF-Complete Bipartite graph. Further we illustrated an example for BFRGS which represents a Route Network system. Moreover, we discussed about Cartesian product of two BFRGS.

1) BF represents Bipolar Fuzzy.
2) BFG represents Bipolar Fuzzy-Graph. 3) F-Graph represents Fuzzy-Graph 4) C-Graph represents Crisp-Graph 5) BFGS represents Bipolar Fuzzy Graph of Semi-group. 6) BF-IGS represents bipolar fuzzy ideal graph of Semi-group. 7) BF-IS represents bipolar fuzzy ideal of a Semi-group 8) SG represents Semi-group. 9) FS represents Fuzzy Subset. 10) BFRG represents Bipolar Fuzzy Regular Graph 11) BFRGS represents Bipolar Fuzzy Regular Graph of a Semi-group 12) 3-PFG represents 3-polar fuzzy graph or tri-polar fuzzy graph 13) 3-PFP represents 3-polar fuzzy Pathor tri-polar fuzzy path 2. Preliminaries Definition 2.1 (Ragamayi, S, 2020) A pair (V, E) is a graph if V ≠ ∅ and E is a set of un-ordered pairs of elements of V.   Definition 2.14 The edge chromatic number of complete bipolar fuzzy graph G = (A, B) on n vertices is (n, n), if n is odd and is (n−1, n−1), if n is even.

Vertex and Edge Chromatic Number of a Bipolar Fuzzy of a Complete Bipartite Graph
In this segment, we propose the concept of BFG signifying a Route network representing a regular Graph of a SG as a generalization of BFG and Regular Graph and C-Graph. Here, we work on simple graphs having limited number of routes (Edges), Nodes(vertices). Also, we describe Vertex and Edge Chromatic Number of a Bipolar Fuzzy of a Complete Bipartite GraphK 1,3 and K 2,3 through examples. Moreover, we discussed about Cartesian product of two BFRGS.

Definition 3.6
The vertex chromatic number of BFGS, Gr (V 1 , A 1 , B 1 , µ, σ ) is (n, n) and is denoted by |χ(Gr)| = (χ P (Gr), χ N (Gr)) = (n, n) where n is the number of vertices of Gr. And The fuzzy value of colouring of a vertex in BFGS is (ℂ P , ℂ N ) where ℂ P , and ℂ N are the fuzzy values of providing and not providing certain color to the vertex.  Also, For a complete bipartite graph K 1,3 Since no two adjacent colours should fill with same color, Hence, the vertex chromatic number of a BFGS of a Complete Bipartite Graph is (2,2).

Labelling of Tri-polar Fuzzy of Graph
Definition 4.1 An edge AB is called a 3-polar fuzzy bridge of G if its removal reduces the strength of connectedness between some other pair of nodes in G.

Definition 4.2
In an edge AB, the node B is stated to be3-polar fuzzy cut node of G if its removal reduces the strength of connectedness between some other pair of nodes in G.

Definition 4.3
In an edge AB, the node A is stated to be 3-polar fuzzy end node of G if it has exactly one strong neighbour in G.

Conclusion
In this article, we presented the concept of BFRG symptomatic of a Route network system on SG. Also, the notion of BFRGS and the concept of 3-PFP and 3-PFG is characterized through some examples.