Finding small regular graphs of girths 6, 8 and 12 as subgraphs of cages

Small k -regular graphs of girth g where g = 6 , 8 , 12 are obtained as subgraphs of minimal cages. More precisely, we obtain ( k , 6 ) -graphs on 2 ( k q − 1 ) vertices, ( k , 8 ) -graphs on 2 k ( q 2 − 1 ) vertices and ( k , 12 ) -graphs on 2 k q 2 ( q 2 − 1 ) , where q is a prime power and k is a positive integer such that q ≥ k ≥ 3 . Some of these graphs have the smallest number of vertices known so far among the regular graphs with girth g = 6 , 8 , 12 .