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The focus of this article lies on the notion of the edge version of doubly resolving sets (EVDRSs) in circulant networks. EVDRSs refer to unique edge subsets that are necessary for identifying individual edges in a network and distinguishing them based on their edge distances to the elements of the EVDRS. The main objectives were to define the minimal size of EVDRSs for circulant networks $ C_n(1, 2) $ and to investigate their basic properties. The systematic research helped to achieve a new understanding of the existence, construction, and characterization of EVDRSs in circulant networks $ C_n(1, 2) $. It is established that the EVDRSs in the circulant network $ C_n(1, 2) $ are finite and are bounded by the order of the network. Among the numerous implications of these findings are those that refer to the design and optimization of distributed sensor networks, improving communication and network protocols, as well as tracking the spread of infectious diseases and epidemics over social networks. The application of the identified methodology helps improve the process of network optimization which contributes to the development of more effective and robust circulant-based structures.

Monitoring and controlling complex networks is of great importance to understand different types of technological and physical systems for source localization. Source localization refers to the process of determining the location or position of a signal source in space based on measurements obtained from multiple sensors. Doubly resolving sets, also known as doubly-resolving arrays, are a particular type of sensor configuration that can enhance the accuracy of source localization. In other words, source localization in a network is equivalent to calculating minimal doubly resolving sets (mDRS) in a network. The concept of the minimal edge version of doubly resolving sets (evDRS) is extension of mDRS. In this article, we take into account the optimization problem of locating the evDRSs for the classes of generalized prism and grid networks. Also, it is demonstrated that the evDRSs for the classes of generalized prisms and grid networks have constant cardinality. This research presents a novel approach with implications for complex network structures such as, network security and communication systems. Furthermore, the findings may have broader implications for diverse fields such as sensor networks, telecommunications, and distributed computing, where prism and grid-like structures are prevalent. The suggested approach may help to improve network optimization and facilitate more robust and reliable grid-based systems.

Presents corrections to the paper, Edge Version of Doubly Resolving Sets for Grid and Generalized Prism Networks.

Graph theory is widely used to analyze the structure models in chemistry, biology, computer science, operations research and sociology. Molecular bonds, species movement between regions, development of computer algorithms, shortest spanning trees in weighted graphs, aircraft scheduling and exploration of diffusion mechanisms are some of these structure models. Let $G = (V_{G}, E_{G})$ be a connected graph, where $V_{G}$ and $E_{G}$ represent the set of vertices and the set of edges respectively. The idea of the edge version of metric dimension is based on the distance of edges in a graph. Let $R_{E_{G}}$ be the smallest set of edges in a connected graph $G$ that forms a basis such that for every pair of edges $e_{1},e_{2}~\in ~E_{G}$ , there exists an edge $e~\in ~R_{E_{G}}$ for which $d_{E_{G}}(e_{1}, e)~\neq ~d_{E_{G}}(e_{2}, e)$ holds. In this paper, we show that the family of circulant graphs $C_{n}(1,2)$ is the family of graphs with constant edge version of metric dimension.

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.