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(kq-1) vertices, (k,8)-graphs on 2k(q^2-1) vertices and (k,12)-graphs on 2kq^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.

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