Calculating the extremal number image Original Research Article

By the extremal number image we denote the maximum number of edges in a graph of order image and girth at least image. The set of such graphs is denoted by image. In 1975, Erdős mentioned the problem of determining extremal numbers image in a graph of order image and girth at least five. In this paper, we consider a generalized version of the problem for any value of girth by using the hybrid simulated annealing and genetic algorithm (HSAGA). Using this algorithm, some new results for image have been obtained. In particular, we generate some graphs of girth image and 8 which in some cases have more edges than corresponding cages. Furthermore, future work will be described regarding the investigation of structural properties of such extremal graphs and the implementation of HSAGA using parallel computing.