User:Tomruen/Small symmetric graphs: Difference between revisions

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| 2 || K2||[[File:1-simplex t0.svg|60px]] || 1 || 1 || (2₁ 1₂) || 2K1|| || 0 || 0 || 2|| [https://www.weddslist.com/rmdb/1graph.php?gr=k2 WL]

| 3 || K3 [[Triangle]]||[[File:2-simplex t0.svg|60px]] || 3 || 2 || (3₂ 3₂) || 3K1|| || 0 || 0 || 6|| [https://www.weddslist.com/rmdb/1graph.php?gr=k3 WL] [https://mathworld.wolfram.com/TriangleGraph.html MW]

|- style=”background-color:#ffffc0″

| 4 || K4 [[N-skeleton|Sk1]] [[tetrahedron]]||[[File:3-simplex t0.svg|60px]] || 6 || 3 || (4₃ 6₂) || 4K1|| || || || 24|| [https://www.weddslist.com/rmdb/1graph.php?gr=k4 WL] [https://mathworld.wolfram.com/TetrahedralGraph.html MW]

|- style=”background-color:#ffffc0″

| 4 || C4 [[square]]||[[File:Regular_polygon_4_annotated.svg|60px]] || 4 || 2 || (4₂ 4₂) || 2K2||[[File:Regular star figure 2(2,1).svg|60px]] || 2 || 1 || 8|| [https://www.weddslist.com/rmdb/1graph.php?gr=c4 WL] [https://mathworld.wolfram.com/SquareGraph.html MW]

| 5 || K5 [[N-skeleton|Sk1]] [[5-cell]]||[[File:4-simplex t0.svg|60px]] || 10 || 4 || (5₄ 10₂) || 5K1|| || 0 || 0 || 120|| [https://www.weddslist.com/rmdb/1graph.php?gr=k5 WL] [https://mathworld.wolfram.com/PentatopeGraph.html MW]

| 5 || C5 [[pentagon]]||[[File:Regular_polygon_5_annotated.svg|60px]][[File:Regular star polygon 5-2.svg|60px]] || 5 || 2 || (5₂ 5₂) || || || || || 10|| [https://www.weddslist.com/rmdb/1graph.php?gr=c5 WL] [https://mathworld.wolfram.com/CycleGraph.html MW]

|- style=”background-color:#ffffc0″

| 6 || K6 [[N-skeleton|Sk1]][[5-simplex]]||[[File:5-simplex t0.svg|60px]] || 15 || 5 || (6₅ 15₂) || 6K1|| || 0 || 0 || 720|| [https://www.weddslist.com/rmdb/1graph.php?gr=k6 WL]

|- style=”background-color:#ffffc0″

| 6 || K2,2,2 [[N-skeleton|Sk1]] [[octahedron]]||[[File:Complex tripartite graph octahedron.svg|60px]] || 12 || 4 || (6₄ 12₂) || 3K2||[[File:Regular star figure 3(2,1).svg|60px]] || 3 || 1 || 48|| [https://www.weddslist.com/rmdb/1graph.php?gr=k2,2,2 WL] [https://mathworld.wolfram.com/OctahedralGraph.html MW]

A symmetric graph (arc-transitive graph) is a graph that is both vertex transitive and edge transitive.

Key

C_n is Cyclic graph.
K_n is Complete graph.
K_n,n is Complete bipartite graph.
- Complement of K_n,n is 2K_n.
K_n,n,… is multipartite graph.
- Complement of K_n,…,n (m-partite) is mK_n.
nG is unconnected set of n G graphs.
nG is n unconnected graphs G, aut(nG) is n! aut(G)ⁿ.
× is tensor product of graphs (Kronecker product).
- G × K₂ is called a bipartite double cover, making 2 copies of G if G is already bipartite.
- C_n × K₂ = 2C_n, n even (bipartite)
- C_n × K₂ = C_2n, n odd (not bipartite)
G(n,k) is generalized Petersen graph, cubic symmetric for (4,1), (5,2), (8,3), (10,2), (10,3), (12,5), (24,5).
- G(4,1) is skeleton of a cube, (8,12).
- G(5,2) is Petersen graph, (10,15).
- G(8,3) is Möbius–Kantor graph, (16,24).
- G(10,2) is Dodecahedral graph, (20,30).
- G(10,3) is Desargues graph, (20,30).
- G(12,5) is Nauru graph, (24,36).

Families

Generalized orthoplex, multipartite graphs
Bipartite					Tripartite			4-partite		5-partite	6-partite
K_2,2	K_3,3	K_4,4	K_5,5	K_6,6	K_2,2,2	K_3,3,3	K_4,4,4	K_2,2,2,2	K_3,3,3,3	K_2,2,2,2,2	K_2,2,2,2,2,2

List

Symmetric graphs with 2 to 12 vertices
v	Graph					Complement graph				Aut	Src
v	Name	Graph	Edges	Degree	Configuration	Name	Graph	Edges	Degree	Aut	Src
2	K₂		1	1	(2₁ 1₂)	2K₁		0	0	2	WL
3	K₃ Triangle		3	2	(3₂ 3₂)	3K₁		0	0	6	WL MW
4	K₄ Sk₁ tetrahedron		6	3	(4₃ 6₂)	4K₁		0	0	24	WL MW
4	C₄ square		4	2	(4₂ 4₂)	2K₂		2	1	8	WL MW
5	K₅ Sk₁ 5-cell		10	4	(5₄ 10₂)	5K₁		0	0	120	WL MW
5	C_{5_pentagon}		5	2	(5₂ 5₂)					10	WL MW
6	K₆ Sk₁ 5-simplex		15	5	(6₅ 15₂)	6K₁		0	0	720	WL
6	K_2,2,2 Sk₁ octahedron		12	4	(6₄ 12₂)	3K₂		3	1	48	WL MW
6	K_3,3		9	3	(6₃ 9₂)	2K₃		6	2	72	WL MW
6	C₆ hexagon		6	2	(6₂ 6₂)			0		12	WL
7	K₇ Sk₁ 6-simplex		21	6	(7₆ 21₂)	7K₁		0	0	5040	WL
7	C₇ heptagon		7	2	(7₂ 7₂)					14	WL MW
8	K₈ Sk₁ 7-simplex		28	7	(8₇ 28₂)	8K₁		0	0	40320	WL
8	K_2,2,2,2 Sk₁ 16-cell		24	6	(8₆ 24₂)	4K₂		4	1	384	WL MW
8						2C₄		8	2	128
8	K_4,4 Ci₈(1,3)		16	4	(8₄ 16₂)	2K₄		12	3	1152	WL MW
8	Q₃= G(4,1) = K₄×K₂ Sk₁ cube		12	3	(8₃ 12₂)					48	WL MW
8	C₈ octagon		8	2	(8₂ 8₂)					16	WL MW
9	K₉ Sk₁ 8-simplex		36	8	(9₈ 36₂)	9K₁		0	0	362880	WL
9	K_3,3,3		27	6	(9₆ 27₂)	3K₃		9	2	1296	WL
9	K₃×K₃Sk₁ 3-3 duoprism		18	4	(9₄ 18₂)					72	WL
9	C₉ Enneagon		9	2	(9₂ 9₂)					18	WL MW
10	K₁₀ Sk₁ 9-simplex		45	9	(10₉ 45₂)	10K₁		0	0	3628800
10	K₅×K₂ Ci₁₀(1,4)		20	4	(10₄ 20₂)					240	WL MW
10	K_2,2,2,2,2 Sk₁ 5-orthoplex		40	8	(10₈ 40₂)	5K₂		5	1	3840
10						2C₅		10	2	200
10	K_5,5		25	5	(10₅ 25₂)	2K₅		20	4	28800	WL MW
10	G(5,2)		15	3	(10₃ 15₂)					120	WL MW
10	C₁₀ pentagon		10	2	(10₂ 10₂)					20	WL MW
11	K₁₁ Sk₁ 10-simplex Sk₁ 11-cell		55	10	(11₁₀ 55₂)	11K₁		0	0	39916800
11	C₁₁ henagon		11	2	(11₂ 11₂)					22	WL MW
12	K₁₂ Sk₁ 11-simplex		66	11	(12₁₁ 66₂)	12K₁		0	0	479001600
12	K_2,2,2,2,2,2 Sk₁ 6-orthoplex		60	10	(12₁₀ 60₂)	6K₂		6	1	46080	WL
12						2C₆		12	2	288
12						3C₄		12	2	3072
12	K_3,3,3,3		54	9	(12₉ 54₂)	4K₃		12	2	31104	WL
12	K_4,4,4		48	8	(12₈ 48₂)	3K₄		18	3	82944	WL
12						2K_3,3		18	3	10368
12						2K_2,2,2		24	4	4608
12	K_6,6		36	6	(12₆ 36₂)	2K₆		30	5	1036800	WL MW
12	K₄×K₃		36	6	(12₆ 36₂)					144
12	K₆×K₂		30	5	(12₅ 30₂)					1440	WL MW
12	Sk₁ icosahedron		30	5	(12₅ 30₂)					120	WL
12	K_2,2,2×K₂ = C₄×K₃		24	4	(12₄ 24₂)					768	WL
12	Sk₁ (cuboctahedron)		24	4	(12₄ 24₂)					48	WL
12	C₁₂		12	2	(12₂ 12₂)					24	WL MW