Improved Lower Bounds for the Orders of Even Girth Cages

The well-known Moore bound M(k, g) serves as a universal lower bound for the order of k-regular graphs of girth g. The excess e of a k-regular graph G of girth g and order n is the difference between its order n and the corresponding Moore bound, e = n −M(k, g). We find infinite families of parameters (k, g), g > 6 and even, for which we show that the excess of any k-regular graph of girth g is larger than 4. This yields new improved lower bounds on the order of k-regular graphs of girth g of smallest possible order; the so-called (k, g)-cages. We also show that the excess of k-regular graphs of girth g can be arbitrarily large for a restricted family of (k, g)-graphs satisfying an additional structural property and large enough k and g.