On the Construction of a Strong Fuzzy Resolving Set of Fuzzy Graphs

Abstract

Assume   is a fuzzy graph with fuzziness on vertex set  and edge set .  Let  be a fuzzy set on  with  and   The representation of    concerning  is an ordered l–tuples , where    for  and   indicates the maximum strength of all paths connecting  and  in  for 1 ≤ j ≤ l.   The “fuzzy resolving set (FRS)”  of  is the subset of    in which the representations of  concerning  are all distinct. The different representations of   are row-matrices that could be linearly independent or linearly dependent on each other. This paper proposes the “strong fuzzy resolving set (SFRS)” of fuzzy graphs, i.e., the FRS of    in which the representations of  concerning  are linearly independent. The strong resolving number (SRN) is the least cardinality of the underlying set of the SFRS. This article also discusses the connection that exists between an SFRS and the FRS of fuzzy graphs. Furthermore, it is proved that the fuzzy labeling graph  of  vertices, whose the underlying graph is a cycle  has the SRN .