Geodesic
On subgraphs in distance-regular graphs
Journal of Algebraic Combinatorics, Tome 1 (1992) no. 4, pp. 353-362.

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Summary: Terwilliger [15] has given the diameter bound $dle$ (s - 1)$( k - 1) + 1$ for distance-regular graphs with girth $2 s$ and valency $k$. We show that the only distance-regular graphs with even girth which reach this bound are the hypercubes and the doubled Odd graphs. Also we improve this bound for bipartite distance-regular graphs. Weichsel [17] conjectures that the only distance-regular subgraphs of a hypercube are the even polygons, the hypercubes and the doubled Odd graphs and proves this in the case of girth 4. We show that the only distance-regular subgraphs of a hypercube with girth 6 are the doubled Odd graphs. If the girth is equal to 8, then its valency is at most 12.
Keywords: distance-regular graph, hypercubes, doubled odd graph, subgraph, uniformly geodetic graph 
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Koolen, J.H. On subgraphs in distance-regular graphs. Journal of Algebraic Combinatorics, Tome 1 (1992) no. 4, pp. 353-362. http://geodesic.mathdoc.fr/item/JAC_1992__1_4_a1/