Geodesic
Embedding arbitrary graphs into strongly regular and distance regular graphs
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 2 (2005), pp. 218-221

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We show that every simple graph on x vertices is an induced subgraph of some strongly regular graph on fewer than $4x^2$ vertices; which, up to a constant factor, coincides with the existing lower bound. We also show that every simple graph on $x$ vertices is an induced subgraph of some distance regular graph of diameter $3$ on fewer than $8x^3$ vertices, and every simple bipartite graph on $x$ vertices is an induced subgraph of some distance regular bipartite graph of diameter $3$ on fewer than $8x^2$ vertices. 
@article{SEMR_2005_2_a13,
     author = {D. G. Fon-Der-Flaass},
     title = {Embedding arbitrary graphs into strongly regular and distance regular graphs},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {218--221},
     publisher = {mathdoc},
     volume = {2},
     year = {2005},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2005_2_a13/}
}
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D. G. Fon-Der-Flaass. Embedding arbitrary graphs into strongly regular and distance regular graphs. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 2 (2005), pp. 218-221. http://geodesic.mathdoc.fr/item/SEMR_2005_2_a13/