Explicit Formulas for the Complexity of Networks Produced by New Duplicating Corona and Cartesian Product

One important algebraic invariant in networks is complexity. This invariant ensures the accuracy and dependability of the network. In this paper, we employ a combinatorial approach to determine the graph’s complexity. A fundamental set of building blocks (basic graphs) will serve as the foundation for all the graphs we investigate, after which we will analyze the individual blocks and the ways in which they are connected. We compute the spectrum and complexity of a number of fundamental graphs and then we employ the novel duplication corona and Cartesian product operations to construct advanced networks from these graphs. Specifically, straightforward formulas are derived for the complexity of the networks created by the new duplicating corona of the regular graphs (prism, diagonal prism, cycle, complete graph, shadow of the cycle, and Petersen graph) with some families of graphs. Furthermore, using Cartesian product operation, evident and specific formulas for the complexity of the prism of the grid graph Gl,κ, the prism of the stacked book graph Bl,s, the diagonal plane prism grid graph, and the prism of the cylindrical graph Cl,κ are derived.