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The idea of a fault-tolerant metric dimension can be used in chemistry to examine a molecule’s structural resistance to the failure of particular atoms or chemical bonds. This is especially important when creating molecules for specific uses, like drug candidates, as they may be exposed to biological systems or the environment, which could cause certain atoms or bonds to change or break. One such is the dialkyltin complex, which has two alkyl groups joined to a tin atom. The nitrogen atom of an N-salicylidene-L-valine molecule is coupled to the alkyl groups in the dialkyltin complex-1 of N-salicylidene-L-valine, making it a useful catalyst for numerous processes in organic synthesis. The purpose of this study is to identify the fault-tolerant metric dimension of the chain in N-salicylidene-L-valine dialkyltin complex-3. [ABSTRACT FROM AUTHOR]

Graphs are useful for analysing the structure models in computer science, operations research, and sociology. The word metric dimension is the basis of the distance function, which has a symmetric property. Moreover, finding the resolving set of a graph is NP-complete, and the possibilities of finding the resolving set are reduced due to the symmetric behaviour of the graph. In this paper, we introduce the idea of the edge-multiset dimension of graphs. A representation of an edge is defined as the multiset of distances between it and the vertices of a set, B ⊆ V (Γ) . If the representation of two different edges is unequal, then B is an edge-multiset resolving a set of Γ. The least possible cardinality of the edge-multiset resolving a set is referred to as the edge-multiset dimension of Γ. This article presents preliminary results, special conditions, and bounds on the edge-multiset dimension of certain graphs. This research provides new insights into structure models in computer science, operations research, and sociology. They could have implications for developing computer algorithms, aircraft scheduling, and species movement between regions. [ABSTRACT FROM AUTHOR]

A biswapped network is an interconnection network in multiprocessor systems. These are networks of devices or processors and connections of these devices or processors. Here network can be expressed in the form of a graph. Devices or processors represent vertices, and the connection of devices or processors represents edges. The subset of the nodes set is called the resolving set if all the representations or codes are different w.r.t. this subset gives the information about the complete network. The minimum cardinality of the resolving set is called the locating number. In this article, we find the locating number of biswapped interconnection networks. In this paper, We compute the exact values of locating numbers for biswapped networks generated by different families of underlying basis networks like path, cycle, power of path, and complete. We have also given the bounds of locating number of any biswapped network generated by any underlying basis network that consists of its clusters. [ABSTRACT FROM AUTHOR]

In this paper, we study resolvability and fault-tolerant resolvability of convex polytopes and related geometric graphs. Imran et al. (2010) [18] raised an open problem asserting that whether or not every family of convex polytope has a constant metric dimension. Raza et al. (2018) [28] studied the fault-tolerant metric dimension of certain families of convex polytopes. They also concluded their study with an open problem asking whether or not every family of convex polytope has a constant fault-tolerant metric dimension. In this paper, we provide negative answers to both of the aforementioned open problems. We answer the first question by constructing a family of convex polytopes with an unbounded metric dimension. By proving a result between resolvability and fault-tolerant resolvability structures of a graph, we show that the family of convex polytope with an unbounded metric dimension also possesses an unbounded fault-tolerant resolvability structure. Moreover, we construct three more infinite families of graphs which are closely related to convex polytopes, having an unbounded metric dimension. [ABSTRACT FROM AUTHOR]