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

Let q be a prime power and g ∈ {6, 8, 12}. In this paper we obtain (q, g)-graphs on 2qg/2−3(q2 − 1) vertices for g = 6, 8, 12 as subgraphs of known (q + 1, g)-cages. We also obtain (q − r, 6)-graphs on 2(q2 − rq − 1) vertices, and (q − r, 8)-graphs on 2(q3−rq2−q+r) vertices where r is a positive integer such that q ≥ r+3. Furthermore, we find (q − 1, 12)-graphs of order 2q2(q3 − q2 − q + 1). Some of these graphs have the smallest number of vertices known so far among the regular graphs with girth g = 6, 8, 12.