On upper bounds of odd girth cages

We give a construction of kk-regular graphs of girth gg using only geometrical and combinatorial properties that appear in any (k;g+1)(k;g+1)-cage, a minimal kk-regular graph of girth g+1g+1. In this construction, g≥5g≥5 and k≥3k≥3 are odd integers, in particular when k−1k−1 is a power of 2 and (g+1)∈{6,8,12}(g+1)∈{6,8,12} we use the structure of generalized polygons. With this construction we obtain upper bounds for the (k;g)(k;g)-cages. Some of these graphs have the smallest number of vertices known so far among the regular graphs with girth g=5,7,11g=5,7,11.