On Multiplicative Circulant Networks of Order Power of Two: Breadth-first Search Tree, Diameter, Distance Spectral Radius, Forwarding Indices, and Some Distance-based Topological Indices
John Rafael M. Antalan \(^{1,2}\) and Francis Joseph H. Campena \(^2\)

\(^1\)Department of Mathematics and Physics, College of Science, Central Luzon State University, Science City of Munoz, Nueva Ecija, Philippines

\(^2\)Department of Mathematics and Statistics, College of Science, De La Salle University, 2401 Taft Ave., Malate, Manila, Philippines

Abstract

Let \(G\) be a group with identity element \(e\), and assume \(S\subseteq G-\{e\}\). Recall that the graph \(\Gamma\) with \(V(\Gamma)=G\) and \(E(\Gamma)=\{\{g,sg\}:g\in G\ \mbox{and}\ s\in S \}\) is the well-known Cayley graph with connection set \(S\). If our Cayley graph is such that \(G=<\mathbb{Z}_n,+_n>\), then we have a circulant network. Now, let \(m>1\) and \(h\geq 0\) be integers. The graph with vertex set \(\{0,1,...,m^h-2,m^h-1\}\) and edge set \(\{\{u,v\}:\mbox{either}\ u+v\equiv s(mod\ m^h )\ \mbox{or}\ u-v\equiv s(mod\ m^h )\} \) where \(s\in S=\{m^0,m^1,...,m^{h-1}\} \) is called multiplicative circulant network of order \(m^h\) and is denoted by \(MC(m^h)\). Thus, multiplicative circulant networks are special type circulant networks where \(G=\mathbb{Z}_{m^h}\) and \(S=\{m^0,m^1,...,m^{h-1}\}\). Multiplicative circulant networks and circulant networks in general are applied in computer network design, telecommunication networking, and distributed computation.

In this study, we provide a method in constructing the breadth-first search tree (BFS tree) of \(MC(2^h)\). We then use the constructed BFS tree to compute for some important network properties of \(MC(2^h)\) such as the diameter, distance spectral radius, forwarding indices, and some distance-based topological indices.

The provided method in constructing the BFS tree for \(MC(2^h)\) in this study paves an easy way in determining the distance spectral radius, forwarding indices, and some distance-based topological indices of \(MC(2^h)\). Finally, the computed diameter for \(MC(2^h)\) using the method presented in this study agrees with the diameter computed by Arno and Wheeler (1993).

Keywords: network, circulant networks, multiplicative circulant networks, breadth-first search tree, diameter, forwarding index, topological index

Topic: MATHEMATICS AND STATISTICS