Novel resolvability parameter of some well-known graphs and exchange properties with applications

The resolvability parameter is an essential component, especially in the context of network research, due to its theoretical and practical significance. Its importance is evident in several applications and outcomes, including social network analysis, network security, facility location and site selection, and effective routing. We introduce a novel resolvability parameter, Fault-Tolerant Mixed Metric Dimension, in this paper, and this defined as let \(R_{m,f}\) be a set that nodes on a graph as both an edge-resolving set and a resolving set. If \(R_{m,f}\) can uniquely represent the graph’s edges and vertices, then it is referred to as a mixed resolving set, and its all subsets cardinality is called the mixed metric dimension. If all of the graph’s vertices and edges are uniquely represented by \(R_{m,f}^{\prime },\) and all subsets of \(R_{m,f}^{\prime }\) with of cardinality one less than \(R_{m,f}\) likewise have unique representations for all of the graph’s vertices and edges, then \(R_{m,f}\) is referred to as a Fault-Tolerant Mixed Resolving Set, and If two such sets \(R_{m,f}^{1}\) and \(R_{m,f}^{2}\) exist such that \(R_{m,f}^{1}\cap R_{m,f}^{2}\ne 0\) then we say that the graph has exchange property. \(R_{m,f}\)’s minimum cardinality is known as its fault-tolerant mixed Metric Dimension. These definitions offer a means of measuring a collection of vertices’ capacity to represent graph structures uniquely, taking fault-tolerant and resolution into account. Furthermore, a problem related to the lab’s system network is also discussed and linked with this topic in this work. Like a lab engineer is embarking on the creation of a new circular lab, intending to establish where and how many devices within it to supply internet with wire to all systems. A solution to this problem is proving this novel topic authenticity.