Narrow Results

Let $\mathrm{\Psi }=\mathrm{\Psi }(𝕍,𝔼)$ be an undirected (i.e., all the edges are bidirectional), simple (i.e., no loops and multiple edges are allowed), and connected (i.e., between every pair of nodes, there exists a path) graph. Let $d_{\mathrm{\Psi }}({\varrho }_i,{\varrho }_j)$ denotes the number of edges in the shortest path or geodesic distance between two vertices ${\varrho }_i,{\varrho }_j\in 𝕍$. The metric dimension (or the location number) of some families of plane graphs have been obtained in [M. Imran, S. A. Bokhary and A. Q. Baig, Families of rotationally-symmetric plane graphs with constant metric dimension, Southeast Asian Bull. Math.36 (2012) 663–675] and an open problem regarding these graphs was raised that: Characterize those families of plane graphs $\mathrm{\Phi }$ which are obtained from the graph $\mathrm{\Psi }$ by adding new edges in $\mathrm{\Psi }$ such that $\beta (\mathrm{\Psi })=\beta (\mathrm{\Phi })$ and $𝕍(\mathrm{\Phi })=𝕍(\mathrm{\Psi })$. In this paper, by answering this problem, we characterize some families of plane graphs ${\mathrm{\Gamma }}_n$, which possesses the radial symmetry and has a constant metric dimension. We also prove that some families of plane graphs which are obtained from the plane graphs, ${\mathrm{\Gamma }}_n$ by the addition of new edges in ${\mathrm{\Gamma }}_n$ have the same metric dimension and vertices set as ${\mathrm{\Gamma }}_n$, and only 3 nodes appropriately selected are sufficient to resolve all the nodes of these families of plane graphs.

Wireless Sensor Networks (WSNs) have come across several things which include collecting data, handling data and distribution for super visioning specific applications such as the services needed, managing anything that occurred naturally, etc. They totally rely on applications. Therefore, the WSNs are classified under major networks. This is very essential. It can be defined as a network of networks that helps in proper flow of data. The main characteristics of WSN include its continuous changes in topologies, connected nodes with several chips and tunic routing protocol. There should be the better utilization of the available resources so that its life span may exceed. There should be an effective usage of available assets to avoid the waste. In our research, we propose a hybrid approach, namely, the Power Control Tree-Based Cluster (PCTBC) to identify the Sybil attacks in WSNs. It employs several stages structured clustering of nodes based on position and identity verification. This approach is utilized for the usage of energy consumption, effectiveness of detecting Sybil attacks inside clusters. The main aspects put into consideration are the efficient routing protocol of the distance between the hops and the energy power remains, where the distance between the hops is being computed using the Received Signal Strength Indication (RSSI) and also the packet transmission can be properly tuned on the basis of the distance. Also, the proposed approach considers the energy consumption for the transmission of the defined packet.