Certain Metrics in m-Polar Fuzzy Graphs

Abstract

In this paper, we present certain metrics in m-polar fuzzy graphs including antipodal m-polar fuzzy graphs and self median m-polar fuzzy graph of given m-polar fuzzy graphs. We illustrate the concepts of metric in m-polar fuzzy graphs by several examples and investigate some of their properties. We discuss some properties of isomorphic m-polar fuzzy graphs. We also describe an application of m-polar fuzzy graph in a road network. 135 New Math. and Nat. Computation 2016.12:135-155. Downloaded from www.worldscientific.com by WEIZMANN INSTITUTE OF SCIENCE on 06/17/16. For personal use only.concisely one from the corresponding one in Ref. 10. The idea behind this is that \multipolar information" (not just bipolar information which correspond to twovalued logic) exists because data for a real world problems are sometimes from n agents ðn ! 2Þ. For example, the exact degree of telecommunication safety of mankind is a point in ½0; 1 n ðn % 7 Â 10 9 Þ because di®erent person has been monitored di®erent times. There are many examples such as truth degrees of a logic formula which are based on n logic implication operators ðn ! 2Þ, similarity degrees of two logic formula which are based on n logic implication operators ðn ! 2Þ, ordering results of a magazine, ordering results of a university and inclusion degrees (accuracy measures, rough measures, approximation qualities, fuzziness measures, and decision preformation evaluations) of a rough set. Kau®mann 12 gave the de¯nition of a fuzzy graph in 1973 which was based on Zadeh's fuzzy relations. 21 Rosenfeld 16 initiated the idea of fuzzy graph in 1975. By examining fuzzy relations between fuzzy sets, Rosenfeld 16 established various graph theoretical concepts. Further remarks on fuzzy graphs were given by Bhattacharya. 9 Several concepts on fuzzy graphs were introduced by Mordeson and Nair. 4 Ahmed and Gani discussed the concepts of perfect fuzzy graph and self median fuzzy graph in Ref. 6. Akram et al. 1-5 introduced many new concepts, including bipolar fuzzy graphs, regular bipolar fuzzy graphs, metric in bipolar fuzzy graphs, antipodal bipolar fuzzy graphs, and bipolar fuzzy digraph in decision support systems. In this paper, we introduce the notion of an antipodal m-polar fuzzy graph and self median m-polar fuzzy graph of a given m-polar fuzzy graph and prove several characterizations theorems of antipodal m-polar fuzzy graph. We discuss isomorphic properties of antipodal m-polar fuzzy graphs. We also present the application of an m-polar fuzzy digraph by considering a road networking model. We have used standard de¯nitions and terminologies in this paper. For other notations, terminologies and applications not mentioned in the paper, the readers are referred to Refs. 7,8,11,[13][14][15][16][17][18][19]. PreliminariesLet V be a non-empty set. LetṼ 2 denotes the collection of all 2-elements subsets of V .A pair G Ã ¼ ðV ; EÞ, where E Ṽ 2 is called a graph. A simple graph is an undirected graph that has no loops or multiple edges. Formally, given a graph G Ã ¼ ðV ; EÞ, t...

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