Bandwidth of WK-recursive networks and its sparse matrix computation

Recursively scalable networks are a concept in computer science and networking where networks are designed in a way that allows them to expand or scale-up seamlessly by repeating a basic structure or pattern. This approach enables the network to grow in size without requiring a complete redesign or reconfiguration. Many innovative recursive architectures in computer science have been suggested in the literature. Among them, WK-recursive networks are notable interconnection networks that remain in their stand with the properties of being parallel, Hamilton-connected, and fault tolerance that are mainly demanded in computer science engineering and intelligent systems. The bandwidth B(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {B}}(G)$$\end{document} of a given graph is minmax{|f(x)-f(y)|:xy is an element of E(G)}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \min \max \{|f(x)-f(y)|:xy \in E(G)\}$$\end{document} of all one-to-one mapping f:V(G)->{1,2,3,& mldr;,n}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f: V(G) \rightarrow \{1,2,3, \ldots , n\} $$\end{document}. This problem is NP-complete for general graphs and even for trees except for the caterpillar, with a hair length of not more than 2. It has also been proved that it remains NP-complete for cyclic caterpillars. Many researchers found some useful lower and upper bounds for this parameter. At the same time, many architectures in the literature fail to give the exact bandwidth using these bounds. The WK-recursive network discussed in this paper obeys the lower bound given in the literature, and finding the exact bandwidth value is very difficult. With this motivation, we investigate the bandwidth of a WK-recursive network. Also, this paper presents two algorithms which run in polynomial time and give the bandwidth of WK-recursive network for both odd and even cases.