$\text{Diameter-3 distance-regular graphs}$

 No. of vertices Intersection Array Graph $8$ $\{3,2,1;1,2,3\}$ $\text{3-cube }Q_3 \cong H(3,2)$ $10$ $\{4,3,1; 1,3,4\}$ $K_{5,5}-I$ $12$ $\{5,4,1; 1,4,5\}$ $K_{6,6}-I$ $12$ $\{5,2,1;1,2,5\}$ $\text{Icosahedron}$ $14$ $\{3,2,2;1,1,3\}$ $\text{Heawood graph (Incidence graph of }PG(2,2))$ $14$ $\{4,3,2;1,2,4\}$ $\text{Distance-3 graph of Heawood graph (Non-incidence graph of }PG(2,2)\text{)}$ $14$ $\{6,5,1; 1,5,6\}$ $K_{7,7}-I$ $15$ $\{4,2,1;1,1,4\}$ $\text{Line graph of Petersen graph}$ $16$ $\{7,6,1; 1,6,7\}$ $K_{8,8}-I$ $18$ $\{8,7,1; 1,7,8\}$ $K_{9,9}-I$ $20$ $\{9,4,1;1,4,9\}$ $J(6,3)$ $20$ $\{9,8,1; 1,8,9\}$ $K_{10,10}-I$ $21$ $\{4,2,2;1,1,2\}$ $\text{Line graph of Heawood graph (Flag graph of }PG(2,2)\text{)}$ $22$ $\{5,4,3;1,2,5\}$ $\text{Incidence graph of biplane on 11 points}$ $22$ $\{6,5,3;1,3,6\}$ $\text{Incidence graph of }(11,6,3)\text{-design}$ $22$ $\{10,9,1; 1,9,10\}$ $K_{11,11}-I$ $24$ $\{7,4,1;1,2,7\}$ $\text{Klein graph}$ $24$ $\{11,10,1; 1,10,11\}$ $K_{12,12}-I$ $26$ $\{4,3,3;1,1,4\}$ $\text{Incidence graph of }PG(2,3)$ $26$ $\{9,8,3;1,6,9\}$ $\text{Incidence graph of }(13,9,3)\text{-design}$ $26$ $\{12,11,1; 1,11,12\}$ $K_{13,13}-I$ $27$ $\{6,4,2;1,2,3\}$ $H(3,3)$ $27$ $\{8,6,1;1,3,8\}$ $GQ(2,4)\text{ minus spread (2 graphs)}$ $28$ $\{13,16,1;1,6,13\}$ $\text{Taylor graph from }P_{13}$ $28$ $\{13,12,1; 1,12,13\}$ $K_{14,14}-I$ $30$ $\{7,6,4;1,3,7\}$ $\text{Incidence graph of }PG(3,2)$ $30$ $\{7,6,4;1,3,7\}$ $\text{Incidence graphs of Hadamard (15,7,3)-designs}$ $30$ $\{8,7,4;1,4,8\}$ $\text{Incidence graph of complement of }PG(3,2)$ $30$ $\{8,7,4;1,4,8\}$ $\text{Incidence graphs of }(15,8,4)\text{-designs }(N=4)$ $30$ $\{14,13,1; 1,13,14\}$ $K_{15,15}-I$ $32$ $\{6,5,4;1,2,6\}$ $\text{Folded 6-cube}$ $32$ $\{6,5,4;1,2,6\}$ $\text{Incidence graphs of biplanes on 16 points}$ $32$ $\{8,7,4;1,4,8\}$ $\text{Incidence graphs of }(16,10,6)\text{-designs}$ $32$ $\{15,6,1; 1,6,15\}$ $\text{Taylor graph from }J(6,2) \cong \text{ Halved 6-cube}$ $32$ $\{15,8,1; 1,8,15\}$ $\text{Taylor graph from }K(6,2)$ $32$ $\{15,14,1; 1,14,15\}$ $K_{16,16}-I$ $34$ $\{16,15,1; 1,15,16\}$ $K_{17,17}-I$ $35$ $\{4,3,3;1,1,2\}$ $\text{Odd graph }O_4$ $35$ $\{12,6,2;1,4,9\}$ $J(7,3)$ $36$ $\{5,4,2;1,1,4\}$ $\text{Sylvester graph}$ $36$ $\{17,8,1;1,8,17\}$ $\text{Taylor graph from }P_{17}$ $38$ $\{9,8,5;1,4,9\}$ $\text{Incidence graphs of Hadamard (19,9,4)-designs}$ $42$ $\{5,4,4;1,1,5\}$ $\text{Incidence graph of PG(2,4)}$ $42$ $\{6,5,1;1,1,6\}$ $2^{nd}\text{ subconstituent of Hoffman-Singleton graph}$ $52$ $\{6,3,3;1,1,2\}$ $\text{Flag graph of }PG(2,3)$ $56$ $\{15,8,3;1,4,9\}$ $J(8,3)$ $56$ $\{27,10,1;1,10,27\}$ $\text{Gosset graph}$ $56$ $\{27,16,1;1,16,27\}$ $\text{Distance-2 graph of Gosset graph}$ $57$ $\{6,5,2;1,1,3\}$ $\text{Perkel graph}$ $62$ $\{6,5,5;1,1,6\}$ $\text{Incidence graph of }PG(2,5)$ $62$ $\{15,14,8;1,7,15\}$ $\text{Incidence graph of }PG(4,2)$ $63$ $\{6,4,4;1,1,3\}$ $\text{Point graphs of }GH(2,2) \text{ and its dual}$ $63$ $\{8,6,1;1,1,8\}$ $\text{Symplectic 7-cover of }K_9$ $64$ $\{9,6,3;1,2,3\}$ $H(3,4)$ $64$ $\{7,6,5;1,2,3\}$ $\text{Folded 7-cube}$ $64$ $\{21,10,3;1,6,15\}$ $\text{Halved 7-cube}$ $65$ $\{10,6,4;1,2,5\}$ $\text{Hall graph}$ $68$ $\{12,10,3;1,3,8\}$ $\text{Doro graph}$ $80$ $\{13,12,9;1,4,13\}$ $\text{Incidence graph of }PG(3,3)$ $84$ $\{18,10,4;1,4,9\}$ $J(9,3)$ $105$ $\{8,4,4;1,1,2\}$ $\text{Flag graph of }PG(2,4)$ $114$ $\{8,7,7;1,1,8\}$ $\text{Incidence graph of }PG(2,7)$ $120$ $\{21,12,5;1,4,9\}$ $J(10,3)$ $125$ $\{12,8,4;1,2,3\}$ $H(3,5)$ $146$ $\{9,8,8;1,1,9\}$ $\text{Incidence graph of }PG(2,8)$ $175$ $\{12,6,5;1,1,4\}$ $\text{Line graph of Hoffman-Singleton graph}$ $182$ $\{10,9,9;1,1,10\}$ $\text{Incidence graph of }PG(2,9)$ $208$ $\{12,10,5;1,1,8\}$ $\text{Unitary graph from P}\Gamma \text{U(3,4)}$ $266$ $\{12,11,11;1,1,12\}$ $\text{Incidence graph of }PG(2,11)$ $288$ $\{66,65,36;1,30,66\}$ $\text{Incidence graphs of Leonard semibiplanes}$ $352$ $\{175,102,1;1,102,175\}$ $\text{Taylor graph from Higman-Sims group (a)}$ $352$ $\{175,72,1;1,72,175\}$ $\text{Taylor graph from Higman-Sims group (b)}$ $352$ $\{50,49,36;1,14,50\}$ $\text{Incidence graph of Higman's symmetric design}$ $364$ $\{12,9,9;1,1,4\}$ $\text{Point graph of }GH(3,3)$ $506$ $\{15,14,12;1,1,9\}$ $\text{Truncated Witt graph}$ $512$ $\{21,20,16;1,2,12\}$ $\text{Coset graph of doubly truncated binary Golay code}$ $525$ $\{20,18,6;1,1,15\}$ $\text{Unitary graph from P}\Gamma \text{U(3,5)}$ $552$ $\{275,112,1;1,112,275\}$ $\text{Taylor graph from }Co_3 \text{ (a)}$ $552$ $\{275,162,1;1,162,275\}$ $\text{Taylor graph from }Co_3 \text{ (b)}$ $729$ $\{24,22,20;1,2,12\}$ $\text{Coset graph of extended ternary Golay code}$ $759$ $\{30,28,24;1,3,15\}$ $\text{Witt graph}$ $1024$ $\{22,21,20;1,2,6\}$ $\text{Coset graph of truncated binary Golay code}$ $1024$ $\{231,160,6;1,48,210\}$ $\text{Distance-2 graph of coset graph of truncated binary Golay code}$ $1395$ $\{98,72,32;1,9,49\}$ $\text{Grassmann graph }J_2(6,3)$

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Last updated: 4 August 2017