Chang graphs

Chang proved that the triangular graphs T(n) are uniquely determined by their parameters (as strongly regular graph), with the exception of T(8). Furthermore, that there are up to isomorphism precisely four strongly regular graphs with parameters v = 28, k = 12, λ = 6, μ =4, namely T(8) and three other graphs, known as the Chang graphs.

The three Chang graphs can be obtained by Seidel switching from T(8) (the line graph of K₈). Namely, switch w.r.t. a set of edges that induces the following subgraph of K₈: (a) 4 pairwise disjoint edges, (b) C₃ + C₅, (c) an 8-cycle C₈.

name group size max cliques max cocliques clique cover chromatic number

T(8) 40320 3⁵⁶,7⁸ 4¹⁰⁵ 6 7

Chang1 384 4³²,5²⁴,6⁸ 3¹²⁸,4⁷³ 7 7

Chang2 360 4⁷⁵,5³⁰,6³ 3¹⁶⁰,4⁶⁵ 8 7

Chang3 96 4⁴⁸,5⁴⁸ 3¹⁶⁰,4⁶⁵ 6 7

name	group size	max cliques	max cocliques	clique cover	chromatic number
T(8)	40320	3⁵⁶,7⁸	4¹⁰⁵	6	7
Chang1	384	4³²,5²⁴,6⁸	3¹²⁸,4⁷³	7	7
Chang2	360	4⁷⁵,5³⁰,6³	3¹⁶⁰,4⁶⁵	8	7
Chang3	96	4⁴⁸,5⁴⁸	3¹⁶⁰,4⁶⁵	6	7

References:

[1] L.C. Chang, The uniqueness and nonuniqueness of triangular association schemes, Sci. Record 3 (1959) 604-613.

[2] L.C. Chang, Association schemes of partially balanced block designs with parameters v = 28, n₁ = 12, n₂ = 15 and p₁₁² = 4, Sci. Record 4 (1960) 12-18.