A new algorithm to find fuzzy Hamilton cycle in a fuzzy network using adjacency matrix and minimum vertex degree.

A Nagoor Gani,S R Latha

doi:10.1186/s40064-016-3473-x

Abstract

A Hamiltonian cycle in a graph is a cycle that visits each node/vertex exactly once. A graph containing a Hamiltonian cycle is called a Hamiltonian graph. There have been several researches to find the number of Hamiltonian cycles of a Hamilton graph. As the number of vertices and edges grow, it becomes very difficult to keep track of all the different ways through which the vertices are connected. Hence, analysis of large graphs can be efficiently done with the assistance of a computer system that interprets graphs as matrices. And, of course, a good and well written algorithm will expedite the analysis even faster. The most convenient way to quickly test whether there is an edge between two vertices is to represent graphs using adjacent matrices. In this paper, a new algorithm is proposed to find fuzzy Hamiltonian cycle using adjacency matrix and the degree of the vertices of a fuzzy graph. A fuzzy graph structure is also modeled to illustrate the proposed algorithms with the selected air network of Indigo airlines.

Highlights

The ideas of Graph theory are highly utilized by computer science applications and research areas such as data mining, image segmentation, clustering, image capturing, networking, etc
Walks, and circuits in graph theory are used in tremendous applications like traveling salesman problem, database design concepts and resource networking. This leads to the development of new algorithms and new theorems
In 2015, Dudek and Ferrara (2015) revised the results proved by Dudek, Frieze, and Rucinski (2012) and showed that there is a constant c′ = c′(k, l) such that every l, cn′k−l —bounded edge—colored Kn(k) contains a properly colored l− overlapping Hamilton cycle

Summary

Background

The ideas of Graph theory are highly utilized by computer science applications and research areas such as data mining, image segmentation, clustering, image capturing, networking, etc. Walks, and circuits in graph theory are used in tremendous applications like traveling salesman problem, database design concepts and resource networking. This leads to the development of new algorithms and new theorems. Chvatal (1972) provided the best possible generalizations of the theorems of Dirac (1952), Posa (1962) and Bondy (1969) that gave successively weaker sufficient conditions for a graph to be Hamiltonian. He deduced some corollaries on Hamiltonian paths, n—Hamiltonian graphs and Hamiltonian bipartite graphs.

The proof uses

Conclusion