Abstract
Fuzzy graph (FG) models embrace the ubiquity of existing in natural and man-made structures, specifically dynamic processes in physical, biological, and social systems. It is exceedingly difficult for an expert to model those problems based on a FG because of the inconsistent and indeterminate information inherent in real-life problems being often uncertain. Vague graph (VG) can deal with the uncertainty associated with the inconsistent and determinate information of any real-world problem, where FGs many fail to reveal satisfactory results. Regularity definitions have been of high significance in the network heterogeneity study, which have implications in networks found across biology, ecology, and economy; so, adjacency sequence (AS) and fundamental sequences (FS) of regular vague graphs (RVGs) are defined with examples. One essential and adequate prerequisite has been ascribed to a VG with maximum four vertices is that it should be regular based on the adjacency sequences concept. Likewise, it is described that if ζ and its principal crisp graph (CG) are regular, then all the nodes do not have to have the similar AS. In the following, we obtain a characterization of vague detour (VD) g-eccentric node, and the concepts of vague detour g-boundary nodes and vague detour g-interior nodes in a VG are examined. Finally, an application of vague detour g-distance in transportation systems is given.
Highlights
Represented Fuzzy graph (FG) based on Zadeh’s fuzzy relation [5, 6]
Ghorai et al [22, 23] defined detour g-interior nodes in bipolar fuzzy graphs and characterization of regular bipolar fuzzy graphs. e idea of strong arcs in FG was given by Bhutani and Rosenfeld [24], and types of arc in FG were given by Mathew and Sunitha [25]. e notion of bridge, trees, cycles, Journal of Mathematics and end nodes were described by Rosenfeld [1]
A Vague graph (VG) is referred to as a generalized structure of an FG that conveys more exactness, adaptability, and compatibility to a system when coordinated with systems running on FGs
Summary
In which the elements are taken from the interval [0, 1]. Definition 1 (see [3]). Ak b in ζ is a sequence of distinct nodes so that tN(ai−1ai) > 0, fN(ai−1ai) > 0, i 1, 2, . K and the length of the path is k. IbfetPw:eean aaa0n, da1b,,.t.h.e,na,k( tNb(abbe))ak path of length k and (fN(ab))k are defined tN(ak−1 as (tN(ab))k b)} and (fsuNp(atNb)()ak a 1i)n∧f tN(a1a2)∧ ··· ∧ fN (aa1)∨. (t∞ N (ab),f∞ N (ab)) is said to be the strength of connectedness between two nodes a and b in ζ, where t∞ N (ab) sup(tkN(ab)) and f∞ N (ab) inf(fkN(ab)). A connected-VG ζ is said to be a vague tree if ζ has a vague spanning subgraph S (A, E) which is a vague tree and for all arcs (a, b) not in S, tN(ab) < t∞ N (ab) and fN(ab) > f∞ N (ab).
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