Fuzzy Tree t -Spanners of Ratio Labelled Butterfly Network

Abstract

Background/Objectives: The reliability and optimality of a communication system depend on the network that has a minimum distance. The minimum distance is the shortest path between the vertices. Even in error correction codes, the minimum Hamming distance ensures data integrity. Hence to have efficient routing protocols, the network is preferred to be a spanning tree that connects all the nodes of the communication system. As tree t- spanner is the optimization technique in determining minimum spanning tree, finding tree t-spanners of a ratio labelled fuzzy graph for the Butterfly network BF(n) is notion of this study. Methods: This study has introduced a new labelling called Ratio Labelling (RL) for examining the fuzziness in a crisp graph. Using the RL method, the crisp graphs were examined for the admittance of fuzziness and were thereby found, to have a tree t-spanners of ratio labelled fuzzy graphs. Findings: The quality of a network can be measured using the stretch factor of the network. The issues of network survival or link failures in a communication networks can be rectified by approximating the best flow spanner. As the optimal connectivity between the nodes is ensured by interconnection networks, the fuzzy tree t- spanners of undirected fuzzy Butterfly Network BF(n) are examined by labelling the vertices and edges using ratio labelling. The Breadth First Search (BFS) algorithm enhances the process of minimizing the spanning tree for an unweighted graph. Since the edge weightage is unique for all the edges of BF(n) under RL, searching for an edge with minimum weight by Kruskal’s algorithm or Prim’s algorithm are less meaningful. Hence, the BFS algorithm is adopted in finding the spanning tree T of Ratio Labelled BF(n). The bounds for the stretch factor t of a fuzzy tree t- spanner of ratio labelled BF(n) were obtained. Novelty: The literature survey shows that various classes of graphs were examined for the admissibility of tree t-spanners. Finding a tree t-spanner for a fuzzy graph was not found in literature, as per our knowledge. Extending the problem of finding a tree t -spanner of a ratio labelled fuzzy graph is an advancement in the field of fuzzy graphs. Keywords: Fuzzy Graph, Fuzzy distance, Ratio Labelling, Tree t- spanner, Butterfly Network