Abstract
Graphs are used as models to solve problems in fields such as mathematics, computer science, physics, and chemistry. In particular, torus, hypercube, and star graphs are popular when modeling the connection structure of processors in parallel computing because they are symmetric and have a low network cost. Whereas a hypercube has a substantially smaller diameter than a torus, star graphs have been presented as an alternative to hypercubes because of their lower network cost. We propose a novel log star (LS) that is symmetric and has a lower network cost than a star graph. The LS is an undirected, recursive, and regular graph. In LSn, the number of nodes is n! while the degree is 2log2n − 1 and the diameter is 0.5n(log2n)2 + 0.75nlog2n. In this study, we analyze the basic topological properties of LS. We prove that LSn is a symmetrical connected graph and analyzed its subgraph characteristics. Then, we propose a routing algorithm and derive the diameter and network cost. Finally, the network costs of the LS and star graph-like networks are compared.
Highlights
Network Cost Than Star Graphs.A graph consists of nodes and edges
Graphs have been applied to the topology of parallel computers, the connection structure of very large-scale integrated (VLSI) internal cores, the connection structure of sink nodes and source nodes in wireless sensor networks (WSNs), and sorting problems
Graphs are designed based on the characteristics of the problem to be solved, which determine the most suitable evaluation measures for a graph
Summary
The evaluation measures of a graph include the number of nodes, number of edges, degree, diameter, network cost, bisection bandwidth, connectivity, symmetry, expandability, and fault tolerance. The network cost of a graph is its degree × diameter. Minimizing the network cost is highly similar to the (d,k) problem and the packing density problem These problems involve finding a graph that has a small degree and diameter. Because there is a trade-off between the degree and diameter, finding a graph with a low network 3cost is Electronics 2021, 10, x FOR PEER REVIEW of 12 difficult [5,6]. A hypercube Qn has a diameter of n, degree in The. LS8, star the (connected edge, neighboring node) for node of n,For andexample, 2n nodes.
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