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Connectivity is a key theory in fuzzy incidence graphs FIGs . In this paper, we introduced connectivity index CI , average connectivity index ACI , and Wiener index WI of FIGs . Three types of nodes including fuzzy incidence connectivity enhancing node FICEN , fuzzy incidence connectivity reducing node FICRN , and fuzzy incidence connectivity neutral node FICNN are also discussed in this paper. A correspondence between WI and CI of a FIG is also computed.

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The number of edges in a shortest walk (without repetition of vertices) from one vertex to another vertex of a connected graph G is known as the distance between them. For a vertex x and an edge e = a b in G , the minimum number from distances of x with a and b is said to be the distance between x and e . A vertex x is said to distinguish (resolves) two distinct edges e 1 and e 2 if the distance between x and e 1 is different from the distance between x and e 2 . A set X of vertices in a connected graph G is an edge metric generator for G if every two edges of G are distinguished by some vertex in X . The number of vertices in such a smallest set X is known as the edge metric dimension of G . In this article, we solve the edge metric dimension problem for certain classes of planar graphs.

There is extremely a great deal of mathematics associated with electrical and electronic engineering. It relies upon what zone of electrical and electronic engineering; for instance, there is much increasingly theoretical mathematics in communication theory, signal processing and networking, and so forth. Systems include hubs speaking with one another. A great deal of PCs connected together structure a system. Mobile phone clients structure a network. Networking includes the investigation of the most ideal method for executing a system. Graph theory has discovered a significant use in this zone of research. In this paper, we stretch out this examination to interconnection systems. Hierarchical interconnection systems (HINs) give a system to planning systems with diminished connection cost by exploiting the area of correspondence that exists in parallel applications. HINs utilize numerous levels. Lower-level systems give nearby correspondence, while more significant level systems encourage remote correspondence. HINs provide issue resilience within the sight of some defective nodes and additionally interfaces. Existing HINs can be comprehensively characterized into two classes: those that use nodes or potential interface replication and those that utilize reserve interface nodes.