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# Paulus graphs

Paulus (1973) made a complete enumeration of the conference matrices of order 25 and the two-graphs of order 26. He finds that up to isomorphism there are 15 strongly regular graphs with parameters v = 25, k = 12, λ = 5, μ = 6 (and spectrum 121 212 (-3)12), and 10 strongly regular graphs with parameters v = 26, k = 10, &lambda = 3, &mu = 4 (and spectrum 101 213 (-3)12).

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.

## Regular two-graphs on 26 vertices

There are 4 regular two-graphs on 26 vertices. Some properties:

name group size children
A 6 16,26,33,43,53,63,7,8
B 72 912,1012,11,12
C 39 1313,1413
D 15600 1526

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.

## Conference graphs on 25 vertices

There are 15 conference graphs on 25 vertices. Some properties:

name group size two-graph complement max cliques comments
P25.01 1 A P25.02 37,474,53
P25.02 1 A P25.01 35,474,53
P25.03 2 A P25.04 38,472,53
P25.04 2 A P25.03 38,472,53
P25.05 2 A P25.06 34,474,53
P25.06 2 A P25.05 38,474,53
P25.07 6 A P25.08 314,468,53
P25.08 6 A P25.07 314,468,53
P25.09 6 B P25.10 354,458,53
P25.10 6 B P25.09 354,458,53
P25.11 72B P25.12 336,464,53
P25.12 72B P25.11 384,44,515 LS(5)
P25.13 3 C P25.14 33,475,53
P25.14 3 C P25.13 31,475,53
P25.15 600 D P25.15 3100,515 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).

## Strongly regular graphs with parameters (26,10,3,4)

There are 10 strongly regular graphs with parameters (26,10,3,4). Some properties:

name group size two-graph max cliques max cocliques comments
P26.01 1 A 3130 4115,576,61
P26.02 2 A 3130 4116,576,61
P26.03 2 A 3122,42 4100,581,61
P26.04 6 A 3122,42 4104,581,61
P26.05 6 A 398,48 4164,524,613 STS(13)
P26.06 4 B 390,410 4136,570,63
P26.07 6 B 382,412 4124,575,63
P26.08 3 C 3126,41 495,581,61
P26.09 39 C 378,413 4104,539,613 STS(13)
P26.10 120 D 390,410 4210,512,613

References:

A. J. L. Paulus, Conference matrices and graphs of order 26, Technische Hogeschool Eindhoven, report WSK 73/06, Eindhoven, 1973, 89 pp.