## $\text{Distance-regular graphs on up to 50 vertices}$

$N=k" \text{denotes that there are } k \text{ non-isomorphic examples of such graphs.}$
 Graph No. of vertices Diameter $\text{Octahedron }J(4,2)$ $6$ $2$ $\text{Cube }Q_3 \cong K_{4,4}-I \cong H(3,2)$ $8$ $3$ $\text{Paley graph }P_9 \cong H(2,3)$ $9$ $2$ $\text{Petersen graph }O_3 = K(5,2)$ $10$ $2$ $J(5,2)$ $10$ $2$ $K_{5,5}-I$ $10$ $3$ $\text{Icosahedron}$ $12$ $3$ $K_{6,6}-I$ $12$ $3$ $\text{Paley graph }P_{13}$ $13$ $2$ $\text{Heawood graph (Incidence graph of }PG(2,2)\text{)}$ $14$ $3$ $\text{Distance-3 graph of Heawood graph}$ $14$ $3$ $K_{7,7}-I$ $14$ $3$ $\text{Line graph of Petersen graph}$ $15$ $3$ $K(6,2)$ $15$ $2$ $J(6,2)$ $15$ $2$ $\text{4-cube }Q_4 \cong H(4,2)$ $16$ $4$ $H(2,4)$ $16$ $2$ $\text{Complement of }H(2,4)$ $16$ $2$ $\text{Shrikhande graph}$ $16$ $2$ $\text{Complement of Shrikhande graph}$ $16$ $2$ $\text{Clebsch graph}$ $16$ $2$ $\text{Complement of Clebsch graph}$ $16$ $2$ $K_{8,8}-I$ $16$ $2$ $\text{Paley graph }P_{17}$ $17$ $2$ $\text{Pappus graph}$ $18$ $4$ $K_{9,9}-I$ $18$ $3$ $\text{Dodecahedron}$ $20$ $5$ $\text{Desargues graph }D(O_3)$ $20$ $5$ $J(6,3)$ $20$ $3$ $K_{10,10}-I$ $20$ $3$ $\text{Line graph of Heawood graph}$ $21$ $3$ $J(7,2)$ $21$ $2$ $K(7,2)$ $21$ $2$ $\text{Incidence graph of biplane on 11 points}$ $22$ $3$ $\text{Incidence graph of }(11,6,3)\text{-design}$ $22$ $3$ $K_{11,11}-I$ $22$ $3$ $\text{Klein graph}$ $24$ $3$ $K_{12,12}-I$ $24$ $3$ $H(2,5)$ $25$ $2$ $\text{Paley graph }P_{25}$ $25$ $2$ $\text{Paulus graphs, SRG(25,12,5,6) }(N=14, \text{ 7 pairs)}$ $25$ $2$ $\text{Complement of }H(2,5)$ $25$ $2$ $\text{Paulus graphs, SRG(26,10,3,4) }(N=10)$ $26$ $2$ $\text{Complements of Paulus graphs, SRG(26,15,8,9) }(N=10)$ $26$ $2$ $\text{Incidence graph of PG(2,3)}$ $26$ $3$ $\text{Incidence graph of }(13,9,3)\text{-design}$ $26$ $3$ $K_{13,13}-I$ $26$ $3$ $\text{Complements of Paulus graphs, srg(26,15,8,9) }(N=10)$ $26$ $2$ $H(3,3)$ $27$ $3$ $GQ(2,4)\text{ minus spread }(N=2)$ $27$ $3$ $\text{Complement of Schl}\ddot{a}\text{fli graph}$ $27$ $2$ $\text{Schl}\ddot{a}\text{fli graph}$ $27$ $2$ $\text{Coxeter graph}$ $28$ $4$ $J(8,2)$ $28$ $2$ $\text{Chang graphs }(N=3)$ $28$ $2$ $\text{Taylor graph from }P_{13}$ $28$ $3$ $K_{14,14}-I$ $28$ $3$ $K(8,2)$ $28$ $2$ $\text{Complements of Chang graphs }(N=3)$ $28$ $2$ $\text{Paley graph }P_{29}$ $29$ $2$ $\text{Other srg(29,14,6,7) }(N=40, 20 \text{ pairs)}$ $29$ $2$ $\text{Tutte's 8-cage}$ $30$ $4$ $\text{Incidence graph of }PG(3,2)$ $30$ $3$ $\text{Incidence graphs of Hadamard (15,7,3)-designs }(N=4)$ $30$ $3$ $\text{Incidence graph of complement of }PG(3,2)$ $30$ $3$ $\text{Incidence graphs of }(15,8,4)\text{-designs }(N=4)$ $30$ $3$ $K_{15,15}-I$ $30$ $3$ $\text{Incidence graph of }AG(2,4) \text{ minus a parallel class}$ $32$ $4$ $\text{5-cube }Q_5 \cong H(5,2)$ $32$ $5$ $\text{Armanios-Wells graph}$ $32$ $4$ $\text{Folded 6-cube}$ $32$ $3$ $\text{Incidence graphs of biplanes on 16 points }(N=3)$ $32$ $3$ $\text{Incidence graphs of }(16,10,6)\text{-designs }(N=3)$ $32$ $3$ $\text{Hadamard graph on 32 vertices}$ $32$ $4$ $\text{Taylor graph from }J(6,2) \cong \text{ Halved 6-cube}$ $32$ $3$ $\text{Taylor graph from }K(6,2)$ $32$ $3$ $K_{16,16}-I$ $32$ $3$ $K_{17,17}-I$ $34$ $3$ $\text{Grassmann graph }J_2(4,2)$ $35$ $2$ $\text{Odd graph }O_4$ $35$ $3$ $J(7,3)$ $35$ $3$ $\text{Sylvester graph}$ $36$ $3$ $\text{Hexacode graph}$ $36$ $4$ $H(2,6)$ $36$ $2$ $\text{Point graphs of }GQ(3,3) \text{ and its dual}$ $40$ $2$ $\text{Coolsaet-Degraer 3-cover of }K_{14}$ $42$ $3$ $\text{Incidence graph of }PG(2,4)$ $42$ $3$ $\text{Symplectic 3-cover of }K_{14}$ $42$ $3$ $2^{\text{nd}} \text{ subconstituent of Hoffman-Singleton graph}$ $42$ $3$ $\text{Line graph of Tutte's 8-cage}$ $45$ $4$ $\text{Halved Foster graph}$ $45$ $4$ $\text{Point graph of }GQ(4,2)$ $45$ $2$ $\text{Hadamard graph on 48 vertices}$ $48$ $4$ $H(2,7)$ $49$ $2$ $\text{Incidence graph of } AG(2,5) \text{ minus a parallel class}$ $50$ $4$ $\text{Hoffman-Singleton graph}$ $50$ $2$ $\text{Complement of Hoffman-Singleton graph}$ $50$ $2$

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Last updated: 20 February 2019