New families of graphs without short cycles and large size Original Research Article

We denote by image or image the maximum number of edges in a graph of order image and girth at least image. First we give a method to transform an image-vertex graph of girth image into a graph of girth at least image on fewer vertices. For an infinite sequence of values of image and image the obtained graphs are denser than the known constructions of graphs of the same girth image. We also give another different construction of dense graphs for an infinite sequence of values of image and image. These two methods improve the known lower bounds on image for image which were obtained using different algorithms. Finally, to know how good are our results, we have proved that image for image, and image for image.