Abstract
Minimum spanning tree finds its huge application in network designing, approximation algorithms for NP-hard problems, clustering problems and many more. Many research works have been done to find minimum spanning tree due to its various applications. But, till date very few research works are available in finding minimum spanning tree in neutrosophic environment. This paper contributes significantly by defining the weight of each network edge using single valued neutro- sophic set (SVNS) and introduce a new approach using similarity measure to find minimum spanning tree in neutrosophic environment. Use of SVNS makes the problem realistic as it can describe the uncertainty, indeterminacy and hesitancy of the real world in a better way. We introduce two new and simple similarity measures to overcome some disadvantages of existing Jaccard, Dice and Cosine similarity measures of SVNSs for ranking the alternatives. Further from the similarity measures we have developed two formulas for the entropy measure proving a fundamental relation between similarity measure and entropy measure. The new entropy measures define the uncertainty more explicitly in comparison to other entropy measure existing in the literature which has been established using an example.
Highlights
A minimum spanning tree of a weighted graph G as discussed by Bang Ye Wu and Kun-Mao Chao in [2] is a spanning tree of G whose edges sum to minimum weight
Further from the similarity measures we have developed two formulas for the entropy measure proving a fundamental relation between similarity measure and entropy measure
We introduce the entropy as a function EN : N(X) → [0, 1] which satisfies the following axioms: (i) EN (A) = 0 if A is a crisp set (ii) EN (A) = 1 if (TA(x), IA(x), FA(x)) = (0.5, 0.5, 0.5) ∀ x ∈ NX (iii) EN (A) ≤ EN (B) if A is less fuzzy than B. i.e
Summary
A minimum spanning tree of a weighted graph G as discussed by Bang Ye Wu and Kun-Mao Chao in [2] is a spanning tree of G whose edges sum to minimum weight. In this paper we at first introduce two new similarity measure functions to overcome some disadvantages of existing Jaccard, Dice and cosine similarity measures of SVNSs discussed in [9] for ranking alternatives Using those new similarity measure formulae, a method to find optimum spanning tree is developed considering the weight of each edge in the graph as SVNS.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
