Closeness Centrality of Cycle and Corona Product Graphs: Application in Fuzzy Transportation Network

Centrality measurement is a useful way to find the most important points (vertices) and connections (edges) in a network‎. ‎Over time‎, ‎many researchers have created different types of centrality measures to study and understand how networks work‎. ‎Closeness centrality‎, ‎in particular‎, ‎is crucial for examining biological‎, ‎social‎, ‎and transportation networks‎. ‎The closeness centrality of a node $u$ of a graph is the multiplicative inverse of the sum of the distances from $u$ to each other vertex‎. ‎We define normalized closeness centrality $C_{NC}(u)$ of a vertex $u$ as $C_{NC}(u) = \frac{n-1}{\sum_{x\in V}\mathfrak{d}(u,x)}$‎, ‎where $n=|V|$‎. ‎This centrality measurement is more receivable than degree centrality because it counts direct as well as indirect connections‎. ‎In this paper‎, ‎we present some new theoretical results for finding normalized closeness centrality of some corona product graphs like $P_{n}\odot P_{m}$‎, ‎$P_{n}\odot K_{m}$‎, ‎$C_{n}\odot K_{m}$‎, ‎$P_{n}\odot C_{m}$‎, ‎$C_{n}\odot C_{m}$‎, ‎$C_{n}\odot P_{m}$‎, ‎$P_{n}\odot S_{m}$‎, ‎$S_{n}\odot K_{m}$‎, ‎$K_{n}\odot P_{m}$ and $K_{n}\odot K_{m}$‎. ‎We also correct the result established by Eballe et al‎. ‎for finding the vertex closeness centrality of the cycle graph $C_{n}$‎. ‎The corona graph has many applications‎, ‎including in signed networks‎, ‎biotechnology‎, ‎chemistry‎, ‎and small-world networks‎. ‎We demonstrate a practical application of our proposed results for identifying influential nodes in small-world networks based on our results and using the corona product graph model‎. ‎We also present the applications of our studied results in a transportation network by transferring our crisp result to fuzzy membership degree and using the corona product graph model‎.