The third issue of The IUP Journal of Computational Mathematics features five research articles that focus on the strength of effectiveness of mathematical modeling, optimization techniques and computational algorithms to pursue research of leading quality. It is a well known fact that `Graceful labeling' has remained an active field of research for over three decades, and the topic is a fascinating one in the world of graph theory and discrete mathematics. Several published works and results are available, yet many unsolved problems and unproven conjectures are to be investigated with the hope that new outcomes would emerge.

The first paper, "Graceful Labeling of Some Classes of Diameter Six and Diameter Seven Trees", by Debdas Mishra and Pratima Panigrahi is an attempt to study Graceful labeling of some new classes of diameter six and diameter seven trees applying the concept of joining isomorphic copies of a tree. All trees up to diameter five are graceful, whether all trees of diameter six and seven are graceful is still an open problem. The authors use the methods of joining two or more isomorphic graceful trees from a larger graceful tree on the graceful diameter four and five trees and form some new classes of graceful diameter six and diameter seven trees. Further, they have proved a theorem-related interlaced labeling to some classes of diameter seven trees.

The knowledge of the traditional relationship between living organisms has been one of the primary goals of evolutionary investigation. Phylogeny, i.e., the evolutionary history of a set of organisms, plays a major role in representing and understanding the relationship among various organisms. In their second paper, "Reconstruction of Phylogenetic Tree of Some Selected Housekeeping Proteins Using Distance Based Methods", Veeky Baths, M Arvind, Uday Kiran and Utpal Roy have developed a phylogenetic tree reconstruction method that detects and reports multiple, topologically distant set of proteins from diverse organisms, which is a generalization of the Neighbor-Joining (NJ) method that affords a more thorough sampling of the solution space by keeping track of multiple partial solutions during its execution. The tree was constructed based on the NJ plot. The paper reveals a brief account of the various algorithms such as Needleman-Wunsch and distance based methods such as NJ algorithms that have been used to analyze various protein sequences, which perform multiple sequence alignment and construct phylogenetic trees to determine their evolutionary relationship. Fourteen different species, starting from fission yeast and nematode to the highest order of chimpanzees and humans, were analyzed with as many as eight housekeeping proteins including the tumor suppressor proteins. The results show net divergence of each species from other species. Major outcomes reveal that humans and chimpanzees are closely related in the proteins like dicer1 and p53, while most other proteins indicate that mouse, chick and humans are closely related in terms of evolution.

The third paper, "Load-Flow Analysis of Radial Distribution Networks", by S Ghosh presents a simple algorithm to solve the load-flow problem of radial distribution networks which computes the branch currents, reducing the complexity in computation and exploiting the graphical feature of the radial distribution networks, and further reducing the data preparation. To demonstrate the effectiveness of the proposed method, a comparison has been done with the help of two examples33-node and 69-node radial distribution networks.

Keane (1994) has designed the `bump' function to examine the performance of constrained optimization techniques, which is considered as a standard benchmark for nonlinear constrained optimization. In the fourth paper, "Minimization of Keane's Bump Function by the Repulsive Particle Swarm and the Differential Evolution Methods", S K Mishra optimizes the Keane's bump function of different dimensions by using Repulsive Particle Swarm (RPS) and the Differential Evaluation (DE) techniques of global optimization. The RPS is concerned with intensive local search ability and DE takes care of the most recent formulation of crossover scheme. Results conclude that the DE method is the most effective optimizer in comparison to other methods.

The homotopy perturbation method is the most powerful tool which not only enables one to handle linear equations, but also nonlinear equations. In the last article, "Application of Homotopy Perturbation Method for Two Coupled Scalar Fields", the authors H Jafari, J Sadeghi, M Zabihi and A R Amani apply the homotopy perturbation method to solve the differential equation of coupled system and thereby conclude that the parameter r plays an important role in motion of particles and could be investigated in a certain range of r. Further, when, r(0, 1) the result will be soliton solutions for both u(x) and v(x). The graph of u(x) when r = 0 is non-solitonic, and the graph is minimal for v(x) when r = 1. The results convey that the Homotopy Perturbation Method (HPM) is a powerful and efficient technique to find exact as well as approximate solutions for nonlinear differential equations. In the present day almost all the real world problems do not possess a closed form analytical solution and hence, researchers and scientists are taking approximate numerical solutions for their problems with suitable accuracy, irrespective of whether their problems have linear and/or non-linear models. So, it is virtually a study of change in the changed circumstances. In this process, optimization techniques play a pivotal role.

-- S K Ghosh
Consulting Editor