Diameter-3 distance-regular graphs

No. of verticesIntersection ArrayGraph
8$\{3,2,1;1,2,3\}$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$\{5,2,1;1,2,5\}$Icosahedron
14$\{3,2,2;1,1,3\}$Heawood graph (Incidence graph of $PG(2,2)$)
14$\{4,3,2;1,2,4\}$Distance-3 graph of Heawood graph (Non-incidence graph of $PG(2,2)$)
14$\{6,5,1; 1,5,6\}$$K_{7,7}-I$
15$\{4,2,1;1,1,4\}$Line graph of Petersen graph
16$\{7,6,1; 1,6,7\}$$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\}$$K_{10,10}-I$
21$\{4,2,2;1,1,2\}$Line graph of Heawood graph (Flag graph of $PG(2,2)$)
22$\{5,4,3;1,2,5\}$Incidence graph of biplane on 11 points
22$\{6,5,3;1,3,6\}$Incidence graph of $(11,6,3)$-design
22$\{10,9,1; 1,9,10\}$$K_{11,11}-I$
24$\{7,4,1;1,2,7\}$Klein graph
24$\{11,10,1; 1,10,11\}$$K_{12,12}-I$
26$\{4,3,3;1,1,4\}$Incidence graph of $PG(2,3)$
26$\{9,8,3;1,6,9\}$Incidence graph of $(13,9,3)$-design
26$\{12,11,1; 1,11,12\}$$K_{13,13}-I$
27$\{6,4,2;1,2,3\}$$H(3,3)$
27$\{8,6,1;1,3,8\}$$GQ(2,4)$ minus spread (2 graphs)
28$\{13,16,1;1,6,13\}$Taylor graph from $P_{13}$
28$\{13,12,1; 1,12,13\}$$K_{14,14}-I$
30$\{7,6,4;1,3,7\}$Incidence graph of $PG(3,2)$
30$\{7,6,4;1,3,7\}$Incidence graphs of Hadamard $(15,7,3)$-designs
30$\{8,7,4;1,4,8\}$Incidence graph of complement of $PG(3,2)$
30$\{8,7,4;1,4,8\}$Incidence graphs of $(15,8,4)$-designs (N=4)
30$\{14,13,1; 1,13,14\}$$K_{15,15}-I$
32$\{6,5,4;1,2,6\}$Folded 6-cube
32$\{6,5,4;1,2,6\}$Incidence graphs of biplanes on 16 points
32$\{8,7,4;1,4,8\}$Incidence graphs of $(16,10,6)$-designs
32$\{15,6,1; 1,6,15\}$Taylor graph from $J(6,2) \cong$ Halved 6-cube
32$\{15,8,1; 1,8,15\}$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$\{4,3,3;1,1,2\}$Odd graph $O_4$
35$\{12,6,2;1,4,9\}$$J(7,3)$
36$\{5,4,2;1,1,4\}$Sylvester graph
36$\{17,8,1;1,8,17\}$Taylor graph from $P_{17}$
38$\{9,8,5;1,4,9\}$Incidence graphs of Hadamard $(19,9,4)$-designs
42$\{13,8,1;1,4,13\}$Coolsaet-Degraer 3-cover of $K_{14}$
42$\{5,4,4;1,1,5\}$Incidence graph of $PG(2,4)$
42$\{13,8,1;1,4,13\}$Symplectic 3-cover of $K_{14}$
42$\{6,5,1;1,1,6\}$2nd subconstituent of Hoffman-Singleton graph
51$\{16,10,1;1,5,16\}$Symplectic 3-cover of $K_{17}$
52$\{6,3,3;1,1,2\}$Flag graph of $PG(2,3)$
56$\{15,8,3;1,4,9\}$$J(8,3)$
56$\{27,10,1;1,10,27\}$Gosset graph
56$\{27,16,1;1,16,27\}$Distance-2 graph of Gosset graph
57$\{6,5,2;1,1,3\}$Perkel graph
60$\{11,8,1;1,2,11\}$Symplectic 5-cover of $K_{12}$
62$\{6,5,5;1,1,6\}$Incidence graph of $PG(2,5)$
62$\{15,14,8;1,7,15\}$Incidence graph of $PG(4,2)$
63$\{6,4,4;1,1,3\}$Point graphs of $GH(2,2)$ and its dual
63$\{8,6,1;1,1,8\}$Symplectic 7-cover of $K_9$
64$\{9,6,3;1,2,3\}$$H(3,4)$
64$\{7,6,5;1,2,3\}$Folded 7-cube
64$\{21,10,3;1,6,15\}$Halved 7-cube
65$\{10,6,4;1,2,5\}$Hall graph
68$\{12,10,3;1,3,8\}$Doro graph
80$\{13,12,9;1,4,13\}$Incidence graph of $PG(3,3)$
84$\{18,10,4;1,4,9\}$$J(9,3)$
85$\{16,12,1;1,3,16\}$Symplectic 5-cover of $K_{17}$
105$\{8,4,4;1,1,2\}$Flag graph of $PG(2,4)$
114$\{8,7,7;1,1,8\}$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)$
135$\{14,12,8;1,3,7\}$Dual polar graph $\mathrm{B}_3(2) \cong$ Dual polar graph $\mathrm{C}_3(2)$
146$\{9,8,8;1,1,9\}$Incidence graph of $PG(2,8)$
165$\{24,14,6;1,4,9\}$$J(11,3)$
175$\{12,6,5;1,1,4\}$Line graph of Hoffman-Singleton graph
182$\{10,9,9;1,1,10\}$Incidence graph of $PG(2,9)$
186$\{10,5,5;1,1,2\}$Flag graph of $PG(2,5)$
208$\{12,10,5;1,1,8\}$Unitary graph from $\mathrm{P} \Gamma \mathrm{U}(3,4)$
216$\{15,10,5;1,2,3\}$$H(3,6)$
220$\{27,16,7;1,4,9\}$$J(12,3)$
266$\{12,11,11;1,1,12\}$Incidence graph of $PG(2,11)$
286$\{30,18,8;1,4,9\}$$J(13,3)$
288$\{66,65,36;1,30,66\}$Incidence graphs of Leonard semibiplanes
343$\{18,12,6;1,2,3\}$$H(3,7)$
352$\{175,102,1;1,102,175\}$Taylor graph from Higman-Sims group (Graph 1)
352$\{175,72,1;1,72,175\}$Taylor graph from Higman-Sims group (Graph 2)
352$\{50,49,36;1,14,50\}$Incidence graph of Higman's symmetric design
364$\{33,20,9;1,4,9\}$$J(14,3)$
364$\{12,9,9;1,1,4\}$Point graph of $GH(3,3)$
455$\{36,22,10;1,4,9\}$$J(15,3)$
462$\{36,25,16;1,4,18\}$Folded Johnson graph $J(12,6)$
506$\{15,14,12;1,1,9\}$Truncated Witt graph
512$\{21,20,16;1,2,12\}$Coset graph of doubly truncated binary Golay code
512$\{21,14,7;1,2,3\}$$H(3,8)$
512$\{49,36,16;1,6,28\}$$\mathrm{H}_2(3,3)$ graph
525$\{20,18,6;1,1,15\}$Unitary graph from $\mathrm{P} \Gamma \mathrm{U}(3,5)$
552$\{275,112,1;1,112,275\}$Taylor graph from $Co_3$ (a)
552$\{275,162,1;1,162,275\}$Taylor graph from $Co_3$ (b)
560$\{39,24,11;1,4,9\}$$J(16,3)$
672$\{110,81,12;1,18,90\}$Moscow-Soicher graph
680$\{42,26,12;1,4,9\}$$J(17,3)$
729$\{24,22,20;1,2,12\}$Coset graph of extended ternary Golay code
729$\{24,16,8;1,2,3\}$$H(3,9)$
759$\{30,28,24;1,3,15\}$Witt graph
765$\{28,24,16;1,3,7\}$Dual polar graph $\phantom{.}^2 \mathrm{D}_4(2)$
816$\{45,28,13;1,4,9\}$$J(18,3)$
891$\{42,40,32;1,5,21\}$Dual polar graph $\phantom{.}^2 \mathrm{A}_6(2)$
969$\{48,30,14;1,4,9\}$$J(19,3)$
1000$\{27,18,9;1,2,3\}$$H(3,10)$
1024$\{22,21,20;1,2,6\}$Coset graph of truncated binary Golay code
1024$\{231,160,6;1,48,210\}$Distance-2 graph of coset graph of truncated binary Golay code
1024$\{66,45,28;1,6,30\}$Halved folded 12-cube
1024$\{33,30,15;1,2,15\}$Shi-Krotov-Solé graph
1120$\{39,36,27;1,4,13\}$Dual polar graph $\mathrm{C}_3(3)$
1140$\{51,32,15;1,4,9\}$$J(20,3)$
1330$\{54,34,16;1,4,9\}$$J(21,3)$
1331$\{30,20,10;1,2,3\}$$H(3,11)$
1395$\{98,72,32;1,9,49\}$Grassmann graph $J_2(6,3)$
1540$\{57,36,17;1,4,9\}$$J(22,3)$
1716$\{49,36,25;1,4,9\}$Folded Johnson graph $J(14,7)$
1728$\{33,22,11;1,2,3\}$$H(3,12)$
1771$\{60,38,18;1,4,9\}$$J(23,3)$
2024$\{63,40,19;1,4,9\}$$J(24,3)$
2048$\{23,22,21;1,2,3\}$Coset graph of binary Golay code
2048$\{253,210,3;1,30,231\}$Distance-2 graph of coset graph of binary Golay code
2197$\{36,24,12;1,2,3\}$$H(3,13)$
2300$\{66,42,20;1,4,9\}$$J(25,3)$
2600$\{69,44,21;1,4,9\}$$J(26,3)$
2744$\{39,26,13;1,2,3\}$$H(3,14)$
2925$\{72,46,22;1,4,9\}$$J(27,3)$
3276$\{75,48,23;1,4,9\}$$J(28,3)$
3375$\{42,28,14;1,2,3\}$$H(3,15)$
3654$\{78,50,24;1,4,9\}$$J(29,3)$
4060$\{81,52,25;1,4,9\}$$J(30,3)$

Back to: Graphs by diameter
Last updated: 12 August 2024