Lower bounds for dilation, wirelength, and edge congestion of embedding graphs into hypercubes

Interconnection networks provide an effective mechanism for exchanging data between processors in a parallel computing system. One of the most efficient interconnection networks is the hypercube due to its structural regularity, potential for parallel computation of various algorithms, and the high degree of fault tolerance. Thus it becomes the first choice of topological structure of parallel processing and computing systems. In this paper, lower bounds for the dilation, wirelength, and edge congestion of an embedding of a graph into a hypercube are proved. Two of these bounds are expressed in terms of the bisection width. Applying these results, the dilation and wirelength of embedding of certain complete multipartite graphs, folded hypercubes, wheels, and specific Cartesian products are computed

Edge Forcing in Butterfly Networks

A zero forcing set is a set $S$ of vertices of a graph $G$, called forcedvertices of $G$, which are able to force the entire graph by applying thefollowing process iteratively: At any particular instance of time, if anyforced vertex has a unique unforced neighbor, it forces that neighbor. In thispaper, we introduce a variant of zero forcing set that induces independentedges and name it as edge-forcing set. The minimum cardinality of anedge-forcing set is called the edge-forcing number. We prove that theedge-forcing problem of determining the edge-forcing number is NP-complete.Further, we study the edge-forcing number of butterfly networks. We obtain alower bound on the edge-forcing number of butterfly networks and prove thatthis bound is tight for butterfly networks of dimensions 2, 3, 4 and 5 andobtain an upper bound for the higher dimensions.Comment: 15 pages, 8 figure

Topological and Thermodynamic Entropy Measures for COVID-19 Pandemic through Graph Theory

Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has caused the global pandemic, coronavirus disease-2019 (COVID-19) which has resulted in 60.4 million infections and 1.42 million deaths worldwide. Mathematical models as an integral part of artificial intelligence are designed for contact tracing, genetic network analysis for uncovering the biological evolution of the virus, understanding the underlying mechanisms of the observed disease dynamics, evaluating mitigation strategies, and predicting the COVID-19 pandemic dynamics. This paper describes mathematical techniques to exploit and understand the progression of the pandemic through a topological characterization of underlying graphs. We have obtained several topological indices for various graphs of biological interest such as pandemic trees, Cayley trees, Christmas trees, and the corona product of Christmas trees and paths. We have also obtained an analytical expression for the thermodynamic entropies of pandemic trees as a function of R0, the reproduction number, and the level of spread, using the nested wreath product groups. Our plots of entropy and logarithms of topological indices of pandemic trees accentuate the underlying severity of COVID-19 over the 1918 Spanish flu pandemic