Analytical and Structural Behaviour of Centrality Measures in Tensor, Corona and Cartesian Graph Products

Graph product operations provide a way to construct complex networks from basic graphs while preserving or modifying selected structural properties. Such constructions can be used to represent systems such as transportation, communication, and distributed computing networks, where the position and importance of vertices depend on the underlying graph structure. In this study, the Tensor, Corona, and Cartesian products obtained from Complete and Wheel graphs are examined. Degree, Closeness, Betweenness, and Eigenvector centralities are considered to study how these graph products affect vertex importance, connectivity, and accessibility. Analytical expressions are developed for the structural properties of the three graph products and are examined through computational experiments using Python and NetworkX. The results show clear differences among the product constructions. In the Tensor product, the degrees of product vertices are determined by the multiplicative degrees of the factor graphs. The Corona product introduces a hierarchical structure through the attachment of Wheel graph copies to the vertices of the Complete graph. In contrast, the Cartesian product produces a comparatively more uniform distribution of structural importance. The numerical results are consistent with the analytical expressions obtained for the considered graph products. The analysis provides a comparative view of how different graph-product constructions influence vertex centrality. An additional Path–Wheel construction is also examined to provide numerical validation beyond the Complete–Wheel setting. The results provide a basis for further study of centrality behaviour in weighted, directed, and dynamic graph-product networks.