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

No. of verticesIntersection ArrayGraph
$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)$

Back to: Graphs by diameter
Last updated: 4 August 2017