These parameter sets are related: a strongly regular graph with parameters (26,10,3,4) is member of the switching class of a regular two-graph, and if one isolates a point by switching, and deletes it, the result is a strongly regular graph with parameters (25,12,5,6). Among these graphs are the Latin square graphs of order 5 on 25 vertices, and the complements of the block graphs of the two Steiner triple systems STS(13) on 26 vertices.
The only one among the graphs on 25 vertices with a transitive group is Paley(25). None of the graphs on 26 vertices has a transitive group.
name | group size | children |
---|---|---|
A | 6 | 1^{6},2^{6},3^{3},4^{3},5^{3},6^{3},7,8 |
B | 72 | 9^{12},10^{12},11,12 |
C | 39 | 13^{13},14^{13} |
D | 15600 | 15^{26} |
Here the 26 children of a regular two-graph on 26 vertices are the serial numbers of the conference graphs on 25 vertices obtained by switching a point isolated.
name | group size | two-graph | complement | max cliques | comments |
---|---|---|---|---|---|
P25.01 | 1 | A | P25.02 | 3^{7},4^{74},5^{3} | |
P25.02 | 1 | A | P25.01 | 3^{5},4^{74},5^{3} | |
P25.03 | 2 | A | P25.04 | 3^{8},4^{72},5^{3} | |
P25.04 | 2 | A | P25.03 | 3^{8},4^{72},5^{3} | |
P25.05 | 2 | A | P25.06 | 3^{4},4^{74},5^{3} | |
P25.06 | 2 | A | P25.05 | 3^{8},4^{74},5^{3} | |
P25.07 | 6 | A | P25.08 | 3^{14},4^{68},5^{3} | |
P25.08 | 6 | A | P25.07 | 3^{14},4^{68},5^{3} | |
P25.09 | 6 | B | P25.10 | 3^{54},4^{58},5^{3} | |
P25.10 | 6 | B | P25.09 | 3^{54},4^{58},5^{3} | |
P25.11 | 72 | B | P25.12 | 3^{36},4^{64},5^{3} | |
P25.12 | 72 | B | P25.11 | 3^{84},4^{4},5^{15} | LS(5) |
P25.13 | 3 | C | P25.14 | 3^{3},4^{75},5^{3} | |
P25.14 | 3 | C | P25.13 | 3^{1},4^{75},5^{3} | |
P25.15 | 600 | D | P25.15 | 3^{100},5^{15} | Paley(25) |
There are two main classes of Latin squares of order 5. One gives Paley(25), the other is labeled here with LS(5).
name | group size | two-graph | max cliques | max cocliques | comments |
---|---|---|---|---|---|
P26.01 | 1 | A | 3^{130} | 4^{115},5^{76},6^{1} | |
P26.02 | 2 | A | 3^{130} | 4^{116},5^{76},6^{1} | |
P26.03 | 2 | A | 3^{122},4^{2} | 4^{100},5^{81},6^{1} | |
P26.04 | 6 | A | 3^{122},4^{2} | 4^{104},5^{81},6^{1} | |
P26.05 | 6 | A | 3^{98},4^{8} | 4^{164},5^{24},6^{13} | STS(13) |
P26.06 | 4 | B | 3^{90},4^{10} | 4^{136},5^{70},6^{3} | |
P26.07 | 6 | B | 3^{82},4^{12} | 4^{124},5^{75},6^{3} | |
P26.08 | 3 | C | 3^{126},4^{1} | 4^{95},5^{81},6^{1} | |
P26.09 | 39 | C | 3^{78},4^{13} | 4^{104},5^{39},6^{13} | STS(13) |
P26.10 | 120 | D | 3^{90},4^{10} | 4^{210},5^{12},6^{13} |
References:
A. J. L. Paulus, Conference matrices and graphs of order 26, Technische Hogeschool Eindhoven, report WSK 73/06, Eindhoven, 1973, 89 pp.
M. Z. Rozenfeld, The construction and properties of certain classes of strongly regular graphs, Uspehi Mat. Nauk 28 (1973) 197-198.
See also Ted Spence's page and the Notebook on Wolfram's page.