Consider an undirected and connected graph $G=(V_G,E_G)$ , where $V_G$ and $E_G$ represent the set of vertices and the set of edges respectively. The concept of edge version of metric dimension and doubly resolving sets is based on the distances of edges in a graph. In this paper, we find the edge version of metric dimension and doubly resolving sets for the necklace graph.

Topological indices are numerical parameters associated with underlying topology of a molecular structure. They are correlated with several physio-chemical properties of chemical compounds. Recently, Euclidean metric based topological index has been introduced named as Sombor index. Therefore, in this article, we will discuss combinatorial aspects and compute Sombor index, average Sombor index and the reduced Sombor index of line graph of silicate carbides Si₂C₃-I[p, q].

In this work, we study the degree-based topological invariants, and the general sum-connectivity, $AB{C}_4$ , $G{A}_5$ , general Zagreb, $GA$ , generalized Randić, and $ABC$ indices of the line graphs of some rooted product graphs ( $C_n\left\{P_k\right\}$ and $C_n\left\{S_{m+1}\right\}$ ) are determined by menas of the concept of subdivision. Moreover, we also computed all these indices of the line graphs of the subdivision graphs of i-th vertex rooted product graph $C_{i,r}\left\{P_{k+1}\right\}$ .

In this paper, we give theoretical results for some topological indices such as Zagreb indices M-1(G), M-2(G), M-3(G), R(G), M-1((G) over bar), M-2((G) over bar), Zagreb co-indices (M-1) over bar (G), (M-2) over bar (G), (M-2) over bar((G) over bar) hyper-Zagreb index HM(G), atom-bond connectivity index ABC(G), sum connectivity index (G) and geometric-arithmetic connectivity index GA(G), by considering G as line graph of subdivision of some convex polytopes and (G) over bar denotes its complement.

Source localization is one of the most challenging problems in complex networks. Monitoring and controlling complex networks is of great interest for understanding different types of systems, such as biological, technological, and complex physical systems. Modern research has made great developments in identifying sensors through which we can monitor or control complex systems. For this task, we choose a set of sensors with the smallest possible size so that the source may be identified. The problem of locating the source of an epidemic in a network is equivalent to the problem of finding the minimal doubly resolving sets (MDRSs) in a network. In this paper, we calculate the minimal doubly resolving sets (MDRSs) of some classes of convex polytopes in order to compute their double metric dimension (DMD).

The concept of resolving sets (RSs) and metric dimension (MD) invariants have a wide range of applications in robot navigation, computer networks, and chemical structure. RS has been used as a sensor in an indoor positioning system to find an interrupter. Many terminologies in machine learning have also been used to diagnose the interrupter in the systems of marine and gas turbines using sensory data. We proposed a fault‐tolerant self‐stable system that allows for the detection of an interrupter even if one of the sensors in the chain fails. If the elimination of any element from a RS is still a RS, then the RS is considered as a fault‐tolerant resolving set (FTRS), and the fault‐tolerant metric dimension (FTMD) is its minimum cardinality. In this paper, we calculated the FTMD of the subdivision graphs of the necklace and prism graphs. We also found that this invariant has constant values for both graphs.