Applications of mathematical programming in graceful labeling of graphs

Graceful labeling is one of the best known labeling methods of graphs. Despite the large number of papers published on the subject of graph labeling, there are few particular techniques to be used by researchers to gracefully label graphs. In this paper, first a new approach based on the mathematical programming technique is presented to model the graceful labeling problem. Then a “branching method” is developed to solve the problem for special classes of graphs. Computational results show the efficiency of the proposed algorithm for different classes of graphs. One of the interesting results of our model is in the class of trees. The largest tree known to be graceful has at most 27 vertices but our model can easily solve the graceful labeling for trees with 40 vertices.

A Mathematical Programming Technique for the Evaluation of Load Distribution and Optimal Modifications for Gear Systems

A generalized technique is presented in this paper for evaluation of load distribution in gear systems and automated selection of optimal modification for the best possible distribution based on any prespecified type of modification. The procedure which utilizes a simplex-type algorithm provides an efficient and effective means for the design of gears with uniform load distribution.