Computation of the Complexity of Networks under Generalized Operations

Abstract

The connected and acyclic components contained in a network are identified by the computation of its complexity, where complexity of a network refers to the total number of spanning trees present within. The article in hand deals with the enumeration of the complexity of various networks’ operations such as sum ( $K_{2,n}+W_3$ , $K_{2,n}+n{K}_1$ , $K_n+S_n$ ), product ( $K_{2,n}\mathrm{⊠}{K}_2$ , $K_{2,n}\mathrm{⋉}{K}_2$ , $K_n\times K_2$ , $K_n\mathrm{⊠}{K}_2$ ), difference $\left(K_{2,n}\mathrm{\ominus }{K}_2\right)$ , and the conjunction of $S_n$ with $K_2$ . All our computations have been concluded by implementation of the methods of linear algebra and matrix theory. Our derivations will also be highlighted with the assistance of 3D plots at the end of this article.