Abstract
Connectivity parameters have an important role in the study of communication networks. Wiener index is such a parameter with several applications in networking, facility location, cryptology, chemistry, and molecular biology, etc. In this paper, we show two notes related to the Wiener index of a fuzzy graph. First, we argue that Theorem 3.10 in the paper “Wiener index of a fuzzy graph and application to illegal immigration networks, Fuzzy Sets and Syst. 384 (2020) 132–147” is not correct. We give a correct statement of Theorem 3.10. Second, by using a new operator on matrix, we propose a simple and polynomial-time algorithm to compute the Wiener index of a fuzzy graph.
Highlights
We discussed two problems related to the Wiener index of a fuzzy graph
We argued that Theorem 3.10 in the paper “Wiener index of a fuzzy graph and application to illegal immigration networks, Fuzzy Sets and Syst. 384 (2020) 132–147” is not correct
We gave a correct statement of Theorem 3.10, where a different result is given for the same conditions
Summary
Elements of σ∗ are called vertices of the fuzzy graph. For any two vertices x and y, let d( x, y) denotes the length of the shortest path between x and y. The strength of connectedness between two vertices x and y is defined as the maximum of the strengths of all paths between x and y and is denoted by ConnG ( x, y). An edge xy of a fuzzy graph G = (σ, μ) is called δ-strong if μ( xy) < ConnG− xy ( x, y).
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