## $\text{Diameter-4 distance-regular graphs}$

 No. of vertices Intersection Array Graph $16$ $\{4,3,2,1; 1,2,3,4\}$ $\text{4-cube }Q_4 \cong H(4,2)$ $18$ $\{3,2,2,1;1,1,2,3\}$ $\text{Pappus graph}$ $28$ $\{3,2,2,1;1,1,1,2\}$ $\text{Coxeter graph}$ $30$ $\{3,2,2,2;1,1,1,3\}$ $\text{Tutte's 8-cage}$ $32$ $\{4,3,3,1;1,1,3,4\}$ $\text{Incidence graph of }AG(2,4) \text{ minus a parallel class}$ $32$ $\{5,4,1,1;1,1,4,5\}$ $\text{Armanios-Wells graph}$ $32$ $\{8,7,4,1;1,4,7,8\}$ $\text{Hadamard graph on 32 vertices}$ $36$ $\{6,5,4,1;1,2,5,6\}$ $\text{Hexacode graph}$ $45$ $\{4,2,2,2;1,1,1,2\}$ $\text{Line graph of Tutte's 8-cage}$ $45$ $\{6,4,2,1;1,1,4,6\}$ $\text{Halved Foster graph}$ $48$ $\{12,11,6,1;1,6,11,12\}$ $\text{Hadamard graph on 48 vertices}$ $50$ $\{5,4,4,1;1,1,4,5\}$ $\text{Incidence graph of }AG(2,5)\text{ minus a parallel class }$ $54$ $\{9,8,6,1;1,3,8,9\}$ $\text{Incidence graph of }\mathrm{STD}_3[9;3]$ $63$ $\{10,6,4,1;1,2,6,10\}$ $\text{Conway-Smith graph}$ $64$ $\{8,7,6,1;1,2,7,8\}$ $\text{Incidence graph of }\mathrm{STD}_2[8;4]$ $72$ $\{12,11,8,1;1,4,11,12\}$ $\text{Suetake graph}$ $80$ $\{4,3,3,3;1,1,1,4\}$ $\text{Incidence graph of }GQ(3,3)$ $81$ $\{8,6,4,2;1,2,3,4\}$ $H(4,3)$ $98$ $\{7,6,6,1;1,1,6,7\}$ $\text{Incidence graph of }AG(2,7) \text{ minus a parallel class}$ $100$ $\{15,14,10,3;1,5,12,15\}$ $\text{Cocliques in Hoffman-Singleton graph}$ $126$ $\{5,4,4,3;1,1,2,2\}$ $\text{Odd graph }O_5$ $128$ $\{8,7,7,1;1,1,7,8\}$ $\text{Incidence graph of }AG(2,8) \text{ minus a parallel class}$ $128$ $\{8,7,6,5;1,2,3,8\}$ $\text{Folded 8-cube}$ $128$ $\{28,15,6,1;1,6,15,28\}$ $\text{Halved 8-cube}$ $162$ $\{6,5,5,4;1,1,2,6\}$ $\text{van Lint-Schrijver graph}$ $162$ $\{9,8,8,1;1,1,8,9\}$ $\text{Incidence graph of }AG(2,9) \text{ minus a parallel class}$ $170$ $\{5,4,4,4;1,1,1,5\}$ $\text{Incidence graph of }GQ(4,4)$ $256$ $\{12,9,6,3;1,2,3,4\}$ $H(4,4)$ $256$ $\{9,8,7,6;1,2,3,4\}$ $\text{Folded 9-cube}$ $256$ $\{36,21,10,3;1,6,15,28\}$ $\text{Halved 9-cube}$ $266$ $\{11,10,6,1;1,1,5,11\}$ $\text{Livingstone graph}$ $280$ $\{9,8,6,3;1,1,3,8\}$ $\text{Unitals in }PG(2,4)$ $288$ $\{12,11,10,7;1,2,5,12\}$ $\text{Leonard graph}$ $315$ $\{10,8,8,2;1,1,4,5\}$ $\text{Hall--Janko/Cohen--Tits near octagon from }J_2.2$ $315$ $\{32,27,8,1;1,4,27,32\}$ $\text{Soicher's } 3^{\text{rd}}\text{ graph}$ $330$ $\{7,6,4,4;1,1,1,6\}$ $\text{Doubly truncated Witt graph}$ $378$ $\{45,32,12,1;1,6,32,45\}$ $\text{Antipodal 3-cover of Zara graph}$ $486$ $\{45,44,36,5;1,9,40,45\}$ $\text{Koolen-Riebeek graph}$ $486$ $\{56,45,16,1;1,8,45,56\}$ $\text{Soicher's } 2^{\text{nd}}\text{ graph}$ $1134$ $\{117,80,24,1;1,12,80,117\}$ $\text{Norton-Smith graph}$ $1344$ $\{176,135,29,1;1,24,135,176\}$ $\text{Meixner double-cover of }U_6(2) \text{ graph}$ $1755$ $\{\}$ $\text{Ree-Tits Generalized Octagon, }GO(2,4)$ $2688$ $\{176,135,36,1;1,12,135,176\}$ $\text{Meixner quadruple cover of }U_6(2) \text{ graph}$ $5346$ $\{416,315,64,1;1,32,315,416\}$ $\text{Soicher's } 1^{\text{st}}\text{ graph}$

Back to: Graphs by diameter
Last updated: 20 March 2019