Abstract
The vague graph (VG), which has recently gained a place in the family of fuzzy graph (FG), has shown good capabilities in the face of problems that cannot be expressed by fuzzy graphs and interval-valued fuzzy graphs. Connectivity index (CI) in graphs is a fundamental issue in fuzzy graph theory that has wide applications in the real world. The previous definitions’ limitations in the connectivity of fuzzy graphs directed us to offer new classifications in vague graph. Hence, in this paper, we investigate connectivity index, average connectivity index, and Randic index in vague graphs with several examples. Also, one of the motives of this research is to introduce some special types of vertices such as vague connectivity enhancing vertex, vague connectivity reducing vertex, and vague connectivity neutral vertex with their properties. Finally, an application of connectivity index in the selected town for building hospital is presented.
Highlights
Introduction efuzzy graph (FG) concept serves as one of the most dominant and extensively employed tools for multiple real-world problem representations, modeling, and analysis
Many of the issues and phenomena around us are associated with complexities and ambiguities that make it difficult to express certainty. ese difficulties were alleviated by the introduction of fuzzy sets by Zadeh [1]. is concept established well-grounded allocation membership degree to elements of a set
Bhattacharya [7] presented some observations on FGs, and some operations on FGs were described by Mordeson and Peng [8]. e existence of a single degree for a true membership could not resolve the ambiguity on uncertain issues, so the need for a degree of membership was felt
Summary
A FG G (V, v, η) is a nonempty set V together with a pair of functions v: V ⟶ [0, 1] and η: V × V ⟶ [0, 1] such that η(ab) ≤ min{v(a), v(b)}, for all a, b ∈ V. (i) A VG G′ (M′, N′) is said to be a PVSG of G if tM′ (a) ≤ tM(a), fM′ (a) ≥ fM(a), ∀a ∈ V and tN′ (ab) ≤ tN(ab), fN′ (ab) ≥ fN(ab), for each edge ab ∈ G. If tN(ab) > 0 and fN(ab) > 0, ∀(a, b) in G, the VG G is called a CVG. (ii) An edge (ab) is said to be a VB if (tN(ab))∞ > (tN′ (ab))∞ and (fN(ab))∞ < (fN′ (ab))∞.
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