Complexity of Some Generalized Operations on Networks

Abstract

The number of spanning trees in a network determines the totality of acyclic and connected components present within. This number is termed as complexity of the network. In this article, we address the closed formulae of the complexity of networks’ operations such as duplication (split, shadow, and vortex networks of $S_n$ ), sum ( $S_n+W_3$ , $S_n+K_2$ , and $\left(C_n\circ K_2\right)+K_1$ ), product ( $S_n\mathrm{⊠}{K}_2$ and $W_n\circ K_2$ ), semitotal networks ( $Q\left(S_n\right)$ and $R\left(S_n\right)$ ), and edge subdivision of the wheel. All our findings in this article have been obtained by applying the methods from linear algebra, matrix theory, and Chebyshev polynomials. Our results shall also be summarized with the help of individual plots and relative comparison at the end of this article.