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The most well-known subject in graph theory is the shortest path problem (SPP), which has real-world applications in several different fields of study, including transportation, emergency services, network communications, fire station services, etc. The arc weights of the applicable SP problems are typically represented by fuzzy numbers in real-world applications. In this paper, we discussed the process of finding the shortest distance in a connected graph network in which the arc weights are multi-valued neutrosophic numbers (MNNs). Moreover, here we compare our method with some of the existing results and illustrate one implementation of our method with the help of one numerical example. [ABSTRACT FROM AUTHOR]

The shortest path (SP) problem (SPP) has several applications in graph theory. It can be used to calculate the distance between the provided initial and final vertex in a network. In this paper, we employed the Fermatean neutrosophic number as the appropriate edge weight of the network to estimate the SP connecting the start and end vertex. This technique is highly useful in establishing the shortest path for the decision-maker under uncertainty. We also investigated its effectiveness in comparison to several existing methods. Finally, a few numerical tests were performed to demonstrate the validity and stability of this new technique, as well as to compare different types of shortest paths with different networks. [ABSTRACT FROM AUTHOR]

We observe that a lobster with diameter at least five has a unique path H = x1, x2, . . . , xm with the property that, besides the adjacency in H, both x1 and xm are adjacent to the centers of at least one K1,s, where s > 0, and each xi, 2 ≤ i ≤ m-1, is at most adjacent to the centers of some K1,s, where s ≥ 0. This unique path H is called the central path of the lobster. We call K1,s an even branch if s is nonzero even, an odd branch if s is odd, and a pendant branch if s = 0. In this paper, we develop methods to get interlaced and graceful lobsters by joining smaller lobsters satisfying certain conditions. The interlaced and graceful lobsters we obtain here may have distinct features than those exist in literature [2], [5], [6], [7], [8], [9], and [11] that (i) some vertices on the central path need not be attached to any non-pendant branch and (ii) some vertices on the central path may have degree odd. We also prove that all diameter four trees other than the caterpillars are not interlaced. [ABSTRACT FROM AUTHOR]