$\text{Incidence graphs}$

$\text{Symmetric designs: projective spaces}$

$\text{Heawood graph}\cong\text{Incidence graph of }PG(2,2)$
$\text{Incidence graph of }PG(2,3)$
$\text{Incidence graph of }PG(2,4)$
$\text{Incidence graph of }PG(2,5)$
$\text{Incidence graph of }PG(2,7)$
$\text{Incidence graph of }PG(2,8)$
$\text{Incidence graph of }PG(2,9)$
$\text{Incidence graph of }PG(2,11)$
$\text{Incidence graph of }PG(3,2)$
$\text{Incidence graph of }PG(3,3)$
$\text{Incidence graph of }PG(4,2)$

$\text{Symmetric designs: biplanes}$

$\text{Distance-3 graph of Heawood graph}\cong\text{Incidence graph of biplane on 7 points}$
$\text{Incidence graph of biplane on 11 points}$
$\text{Incidence graphs of biplanes on 16 points}$

$\text{Symmetric designs: Hadamard designs}$

$\text{Heawood graph}\cong\text{Incidence graph of Hadamard (7,3,1)-design}$
$\text{Incidence graph of Hadamard (11,5,2)-design}$
$\text{Incidence graphs of Hadamard (15,7,3)-designs (3 graphs)}$
$\text{Incidence graphs of Hadamard (19,9,4)-designs (6 graphs)}$

$\text{Other symmetric designs}$

$\text{Incidence graph of }(11,6,3)\text{-design}$
$\text{Incidence graph of }(13,9,3)\text{-design}$
$\text{Incidence graph of complement of }PG(3,2)$
$\text{Incidence graphs of }(15,8,4)\text{-designs }(N=4)$
$\text{Incidence graphs of }(16,10,6)\text{-designs}$
$\text{Incidence graph of Higman's symmetric design}$
$\text{Incidence graph of Leonard semibiplanes}$

$\text{Symmetric transversal designs: affine planes minus a parallel class, i.e. }\mathrm{STD}_1[q;q]$

$\text{Incidence graph of }AG(2,3) \text{ minus a parallel class } \cong \text{ Pappus graph}$
$\text{Incidence graph of }AG(2,4) \text{ minus a parallel class}$
$\text{Incidence graph of }AG(2,5) \text{ minus a parallel class}$
$\text{Incidence graph of }AG(2,7) \text{ minus a parallel class}$
$\text{Incidence graph of }AG(2,8) \text{ minus a parallel class}$
$\text{Incidence graph of }AG(2,9) \text{ minus a parallel class}$

$\text{Symmetric transversal designs: }\mathrm{STD}_\lambda[2\lambda;2]\text{, i.e. Hadamard graphs}$

$\text{Hadamard graph on 16 vertices}\cong 4\text{-cube}\cong\text{Incidence graph of }\mathrm{STD}_2[4;2]$
$\text{Hadamard graph on 32 vertices}\cong\text{Incidence graph of }\mathrm{STD}_4[8;2]$
$\text{Hadamard graph on 48 vertices}\cong\text{Incidence graph of }\mathrm{STD}_6[12;2]$

$\text{Other symmetric transversal designs}$

$\text{Hexacode graph}\cong\text{Incidence graph of }\mathrm{STD}_2[6;3]$
$\text{Incidence graph of }\mathrm{STD}_2[8;4]$
$\text{Incidence graph of }\mathrm{STD}_3[9;3]$
$\text{Suetake graph}\cong\text{Incidence graph of }\mathrm{STD}_4[12;3]$

$\text{Generalized polygons}$

$\text{Incidence graph of }GH(2,2) \cong \text{Tutte's 12-cage}$
$\text{Incidence graph of }GH(3,3)$
$\text{Incidence graph of }GQ(3,3)$
$\text{Incidence graph of }GQ(4,4)$

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Last updated: 7 June 2017