The tables on this web page, and the files linked to, and the two references given at the end, describe the (0,2)-graphs of valency at most 8.

Semibiplanes

A semibiplane is a connected bipartite graph such that any two vertices have 0 or 2 common neighbours. We use v for the number of vertices.

The number of semibiplanes of valency (block size) k at most 8 is
k 0 1 2 3 4 5 6 7 8
N(k) 1 1 1 1 2 4 13 40 104

These semibiplanes are given explicitly below.

A semibiplane can be regarded as a connected point-block incidence structure, where two points are in 0 or 2 blocks, and two blocks meet in 0 or 2 points. A biplane is the particular case where 0 does not occur. In other words, a biplane is a square 2-(w, k, 2) design, where w is the number of points. One has w = k(k-1)/2 + 1, that is, v = 2w = k2-k+2. The known cases have k = 1, 2, 3, 4, 5, 6, 9, 11, 13. See this survey by Gordon Royle, and a description of the three 2-(16,6,2) biplanes.

Rectagraphs

A rectagraph is a connected graph without triangles such that any two vertices have 0 or 2 common neighbours.

Of course all semibiplanes are rectagraphs. The number of nonbipartite rectagraphs of valency k at most 8 is
k 0 1 2 3 4 5 6 7 8
N(k) 0 0 0 0 0 1 2 5 20

These 28 nonbipartite rectagraphs are the graphs N5.2, N6.6, N6.9, N7.44, N7.45, N7.51, N7.52, N7.53, N8.9, N8.37, N8.47, N8.107, N8.108, N8.112, N8.128, N8.131, N8.140, N8.155, N8.158, N8.170, N8.171, N8.178, N8.179, N8.180, N8.188, N8.193, N8.194, N8.195 below.

(0,2)-graphs

A (0,2)-graph is a connected graph such that any two vertices have 0 or 2 common neighbours.

The total number of (0,2)-graphs of valency (block size) k at most 8 is
k 0 1 2 3 4 5 6 7 8
N(k) 1 1 1 2 3 8 24 96 302

These (0,2)-graphs are given explicitly below.

Flag-transitivity

Of the 438 (0,2)-graphs of valency at most 8, 150 have a transitive group of automorphisms (65 bipartite and 85 nonbipartite), and transitivity is indicated in the tables below. Among these 150 there are 31 graphs that are flag-transitive, (23 bipartite and 8 nonbipartite) namely the 9 hypercubes 2k with k in 0..8, the 5 folded hypercubes 2k/1 with k in 3,5,6,7,8, and 17 others, namely 4.1, 5.1, 5.2, 6.1, 6.5, 7.10, 7.29, 8.2, 8.13, 8.19, 8.62, 8.102, N5.1, N6.1, N6.2, N7.12, N8.9.

Chromatic number

Of course bipartite (0,2)-graphs on more than 1 vertex have chromatic number 2. The non-bipartite (0,2)-graphs of valency at most 8 have chromatic number 4, with ten exceptions that have chromatic number 5, namely N7.2, N7.3, N8.1, N8.2, N8.3, N8.5, N8.8, N8.39, N8.44. N8.85. Thus, no (0,2)-graph of valency at most 8 has chromatic number 3.

p-Ranks

The p-ranks, for various primes p, give numerical invariants that sometimes allow to distinguish similar graphs.

Let A be the adjacency matrix. For p=2 we see that any two distinct rows are orthogonal. It follows that A has full 2-rank when k is odd, and 2-rank at most v/2 when k is even. On the other hand, A+I has full 2-rank when k is even and 2-rank at most v/2 when k is odd. In the tables below we give the interesting 2-rank: that of A when k is even, of A+I otherwise. This value is at most v/2, and in 312 of the 438 cases we have equality. (There will be equality for example when k is odd and the graph is bipartite.)

For example, the three biplanes 2-(16,6,2) are distinguished by their 2-ranks 6, 7, 8 - for the incidence graphs this means that the 2-ranks are 12, 14, 16.

One can use 3-ranks, e.g. to distinguish the pairs 8.3-8.5, 8.4-8.6, 8.20-8.21:
  8.3 8.5 8.4 8.6 8.20 8.21
3-rk(A) 66 66 60 64 80 78
3-rk(A+I) 64 66 62 67 81 79

Constructions

Bipartite double

The bipartite double (bipd) of a graph is the product with K2, where two vertices in the product are adjacent when both coordinates are adjacent. The bipartite double of a (0,2)-graph is a bipartite (0,2)-graph of the same valency.

The extended bipartite double (ebipd) of a graph is the graph obtained from the bipartite double by adding the matching consisting of the edges joining two vertices with the same first coordinate. If the original graph is a rectagraph of valency k, the extended bipartite double is a bipartite (0,2)-graph with valency k+1.

Cartesian products

The Cartesian product of (0,2)-graphs is a (0,2)-graph again. In particular, the hypercube 2k is a (0,2)-graph.

Folding

When being at maximal distance is an equivalence relation, the folded graph is the quotient w.r.t. this equivalence relation.

Quotients of projective planes

Given a projective plane Π with involution σ, one obtains a semibiplane of which the points and blocks are the σ-orbits of size 2 on the points and lines of Π (Hughes). If Π has order q, then the incidence graph of the semibiplane has k = q, and one has one of (i) v = q2 (elation, q even); (ii) v = q2-1 (homology, q odd); (iii) v = q2-sqrt(q) (Baer involution, q square). This graph is denoted Π/σ.

Bipartite (0,2)-graphs from root systems

One obtains a bipartite (0,2)-graph given a root system with singly laced diagram and a target vector. Details on a separate page.

Bipartite (0,2)-graphs of valency 0-8

These are the semibiplanes of block size at most 8.

The subgraphs column gives the type and number of the proper (0,2)-subgraphs of largest valency (at least 3), if any.

The 2-rank is that of the adjacency matrix A when k is even, and that of A+I when k is odd.

#kvd |G|orbitsgraphsubgraphs 2-rkspectrum
0.1 0 1 0 1 tra 20   0 0
1.1 1 2 1 2 tra 21   1 1, -1
2.1 2 4 2 8 tra 22   2 2, 0, 0, -2
3.1 3 8 3 48 tra 23   4 3, 1, 1, ... (integral)
4.1 4 14 3 336 tra 2-(7,4,2) biplane   6 4, 1.414, 1.414, ...
4.2 4 16 4 384 tra 24 8 x 3.1 8 4, 2, 2, ... (integral)
5.1 5 22 3 1320 tra 2-(11,5,2) biplane   11 5, 1.732, 1.732, ...
5.2 5 24 4 480 tra bipd(icosahedron)   12 5, 2.236, 2.236, ...
5.3 5 28 4 672 tra 4.1 x K2 2 x 4.1 14 5, 3, 2.414, ...
5.4 5 32 5 3840 tra 25 10 x 4.2 16 5, 3, 3, ... (integral)
6.1 6 32 3 1536 tra 2-(16,6,2) biplane 6 x 4.2 14 6, 2, 2, ... (integral)
6.2 6 32 3 768 tra 2-(16,6,2) biplane 2 x 4.2 16 6, 2, 2, ... (integral)
6.3 6 32 3 23040 tra folded 26
2-(16,6,2) biplane		 30 x 4.2 12 6, 2, 2, ... (integral)
6.4 6 36 4 96 12+24     16 6, 2.828, 2.828, ...
6.5 6 36 4 4320 tra drg(6,5,4,1; 1,2,5,6)   14 6, 2.449, 2.449, ...
6.6 6 36 4 48 12+24     18 6, 2.828, 2.828, ...
6.7 6 40 4 48 8+8+24   9 x 3.1 20 6, 3.020, 3.020, ...
6.8 6 40 4 120 tra   5 x 3.1 20 6, 3.236, 3.236, ...
6.9 6 44 4 2640 tra 5.1 x K2 2 x 5.1 22 6, 4, 2.732, ...
6.10 6 48 5 256 16+32   8 x 4.1 24 6, 3.464, 3.464, ...
6.11 6 48 5 960 tra 5.2 x K2 2 x 5.2 24 6, 4, 3.236, ...
6.12 6 56 5 2688 tra 5.3 x K2 4 x 5.3 28 6, 4, 4, ...
6.13 6 64 6 46080 tra 26 12 x 5.4 32 6, 4, 4, ... (integral)
7.1 7 48 4 384 tra   2 x 5.2 24 7, 3, 3, ...
7.2 7 48 4 1920 tra ebipd(N6.6) 2 x 5.2 24 7, 3, 3, ...
7.3 7 48 4 48 tra   66 x 3.1 24 7, 3, 2.977, ...
7.4 7 48 4 64 16+32   2 x 4.2 24 7, 3, 3, ...
7.5 7 48 4 64 16+32   2 x 4.2 24 7, 3, 3, ...
7.6 7 48 4 48 tra   96 x 3.1 24 7, 3, 3, ...
7.7 7 48 4 96 tra   60 x 3.1 24 7, 3, 3, ...
7.8 7 48 4 72 12+36   84 x 3.1 24 7, 3, 3, ...
7.9 7 48 4 96 tra   6 x 4.2 24 7, 3, 3, ...
7.10 7 48 4 2016 tra Π/σ 84 x 3.1 24 7, 2.646, 2.646, ...
7.11 7 56 4 16 2*4+2*8+2*16   1 x 3.1 28 7, 3.462, 3.352, ...
7.12 7 56 4 8 4*4+5*8   9 x 3.1 28 7, 3.582, 3.557, ...
7.13 7 56 4 16 2*4+2*8+2*16   9 x 3.1 28 7, 3.606, 3.606, ...
7.14 7 56 4 24 2+6+4*12   9 x 3.1 28 7, 3.462, 3.462, ...
7.15 7 56 4 8 4*4+5*8   5 x 3.1 28 7, 3.534, 3.518, ...
7.16 7 56 4 4 14*4   5 x 3.1 28 7, 3.549, 3.540, ...
7.17 7 56 4 16 3*8+2*16   4 x 3.1 28 7, 3.494, 3.494, ...
7.18 7 64 4 48 16+48   56 x 3.1 32 7, 3.711, 3.711, ...
7.19 7 64 4 120 24+40   35 x 3.1 32 7, 3.692, 3.692, ...
7.20 7 64 4 3072 tra 6.1 x K2 2 x 6.1 32 7, 5, 3, ... (integral)
7.21 7 64 4 1536 tra 6.2 x K2 2 x 6.2 32 7, 5, 3, ... (integral)
7.22 7 64 4 1152 tra   8 x 4.2 32 7, 3.606, 3.606, ...
7.23 7 64 4 46080 tra 6.3 x K2 2 x 6.3 32 7, 5, 3, ... (integral)
7.24 7 64 5 12 4+5*12   2 x 4.1 32 7, 3.739, 3.684, ...
7.25 7 64 5 24 4+5*12   38 x 3.1 32 7, 3.988, 3.988, ...
7.26 7 64 5 8 8*4+4*8   33 x 3.1 32 7, 3.952, 3.904, ...
7.27 7 64 5 12 2*2+2*6+4*12   41 x 3.1 32 7, 3.828, 3.638, ...
7.28 7 64 5 336 8+56   56 x 3.1 32 7, 3.828, 3.828, ...
7.29 7 72 5 12096 tra U3(3).2/L2(7) 36 x 4.1 36 7, 3.606, 3.606, ...
7.30 7 72 5 192 24+48 6.4 x K2 2 x 6.4 36 7, 5, 3.828, ...
7.31 7 72 5 8640 tra 6.5 x K2 2 x 6.5 36 7, 5, 3.449, ...
7.32 7 72 5 96 24+48 6.6 x K2 2 x 6.6 36 7, 5, 3.828, ...
7.33 7 80 5 48 8+24+48   1 x 5.2 40 7, 4.236, 4.017, ...
7.34 7 80 5 96 16+16+48 6.7 x K2 2 x 6.7 40 7, 5, 4.020, ...
7.35 7 80 5 240 tra 6.8 x K2 2 x 6.8 40 7, 5, 4.236, ...
7.36 7 88 5 10560 tra 6.9 x K2 4 x 6.9 44 7, 5, 5, ...
7.37 7 96 6 512 32+64 6.10 x K2 2 x 6.10 48 7, 5, 4.464, ...
7.38 7 96 6 3840 tra 6.11 x K2 4 x 6.11 48 7, 5, 5, ...
7.39 7 112 6 16128 tra 6.12 x K2 6 x 6.12 56 7, 5, 5, ...
7.40 7 128 7 645120 tra 27 14 x 6.13 64 7, 5, 5, ... (integral)
8.1 8 64 4 384 tra     32 8, 3.464, 3.464, ...
8.2 8 64 4 10752 tra drg(8,7,6,1; 1,2,7,8)
Π/σ		   26 8, 2.828, 2.828, ...
8.3 8 68 4 4 17*4   118 x 3.1 32 8, 3.708, 3.611, ...
8.4 8 68 4 20 2*4+3*20   131 x 3.1 32 8, 3.689, 3.689, ...
8.5 8 68 4 4 17*4   103 x 3.1 32 8, 3.755, 3.618, ...
8.6 8 68 4 20 2*4+3*20   111 x 3.1 32 8, 3.742, 3.742, ...
8.7 8 72 4 144 tra   90 x 3.1 32 8, 4, 4, ...
8.8 8 74 5 672 2+16+56   126 x 3.1 28 8, 3.646, 3.646, ...
8.9 8 80 4 4 8*2+16*4   52 x 3.1 40 8, 4.255, 4.184, ...
8.10 8 80 4 20 4*20   40 x 3.1 40 8, 4.606, 4.218, ...
8.11 8 80 4 20 4*20   65 x 3.1 36 8, 4.512, 4.179, ...
8.12 8 80 4 384 32+48   32 x 3.1 36 8, 4, 4, ...
8.13 8 80 5 3840 tra   80 x 3.1 32 8, 4, 4, ... (integral)
8.14 8 80 5 12 2*4+6*12   12 x 3.1 40 8, 4.234, 4.149, ...
8.15 8 80 5 768 16+64   16 x 3.1 36 8, 4, 4, ...
8.16 8 84 4 16 4+6*8+2*16   16 x 3.1 40 8, 4.102, 3.907, ...
8.17 8 84 4 64 4+16+2*32   32 x 3.1 40 8, 4.337, 4.337, ...
8.18 8 84 4 32 4+2*8+2*16+32   16 x 3.1 40 8, 4.218, 4.218, ...
8.19 8 84 4 1344 tra 4.L3(2).2/D16   40 8, 3.742, 3.742, ...
8.20 8 84 5 8 7*4+7*8   32 x 3.1 40 8, 4.243, 4.243, ...
8.21 8 84 5 8 7*4+7*8   74 x 3.1 40 8, 4.321, 4.228, ...
8.22 8 84 5 4 14*2+14*4   64 x 3.1 40 8, 4.296, 4.286, ...
8.23 8 84 5 4 14*2+14*4   38 x 3.1 40 8, 4.197, 4.129, ...
8.24 8 84 5 16 4+6*8+2*16   40 x 3.1 40 8, 4.256, 4.218, ...
8.25 8 84 5 16 3*4+3*8+3*16   40 x 3.1 40 8, 4.337, 4.313, ...
8.26 8 84 5 64 4+16+2*32   72 x 3.1 40 8, 4.337, 4.337, ...
8.27 8 84 5 16 3*4+3*8+3*16   40 x 3.1 40 8, 4.337, 4.307, ...
8.28 8 84 5 64 4+16+2*32   24 x 3.1 40 8, 4.307, 4.307, ...
8.29 8 84 5 96 12+24+48   48 x 3.1 40 8, 4.218, 4.218, ...
8.30 8 88 4 160 8+40+40   5 x 4.2 40 8, 4.813, 4.813, ...
8.31 8 88 5 16 3*8+4*16   1 x 4.2 40 8, 4.812, 4.798, ...
8.32 8 88 5 32 3*8+2*32   1 x 4.2 40 8, 4.946, 4.606, ...
8.33 8 96 4 1152 tra   4 x 5.2 48 8, 4, 4, ...
8.34 8 96 5 5760 tra ebipd(N7.44) 6 x 6.11 48 8, 5.236, 5.236, ...
8.35 8 96 5 384 32+64   4 x 5.2 48 8, 5.236, 5.236, ...
8.36 8 96 5 24 2*12+3*24   96 x 3.1 48 8, 4.494, 4.417, ...
8.37 8 96 5 768 tra 7.1 x K2 2 x 7.1 48 8, 6, 4, ...
8.38 8 96 5 3840 tra 7.2 x K2 2 x 7.2 48 8, 6, 4, ...
8.39 8 96 5 192 tra 7.9 x K2 2 x 7.9 48 8, 6, 4, ...
8.40 8 96 5 128 32+64 7.4 x K2 2 x 7.4 48 8, 6, 4, ...
8.41 8 96 5 128 32+64 7.5 x K2 2 x 7.5 48 8, 6, 4, ...
8.42 8 96 5 96 tra 7.6 x K2 2 x 7.6 48 8, 6, 4, ...
8.43 8 96 5 144 24+72 7.8 x K2 2 x 7.8 48 8, 6, 4, ...
8.44 8 96 5 192 tra 7.7 x K2 2 x 7.7 48 8, 6, 4, ...
8.45 8 96 5 96 tra 7.3 x K2 2 x 7.3 48 8, 6, 4, ...
8.46 8 96 5 128 32+64   16 x 4.1
6 x 4.2		 48 8, 5.236, 5.236, ...
8.47 8 96 5 12 12*6+2*12   102 x 3.1 48 8, 4.449, 4.449, ...
8.48 8 96 5 24 8*12   102 x 3.1 46 8, 4.472, 4.366, ...
8.49 8 96 5 48 4*24   78 x 3.1 46 8, 4.472, 4.268, ...
8.50 8 96 5 96 48+48   126 x 3.1 44 8, 4.429, 4.429, ...
8.51 8 96 5 72 24+36+36   96 x 3.1 48 8, 4.415, 4.415, ...
8.52 8 96 5 4032 tra 7.10 x K2 2 x 7.10 48 8, 6, 3.646, ...
8.53 8 100 5 2 28*1+36*2   5 x 4.1 48 8, 4.605, 4.584, ...
8.54 8 100 5 4 25*4   2 x 4.1 48 8, 4.643, 4.528, ...
8.55 8 100 5 2 22*1+39*2   4 x 4.1 48 8, 4.497, 4.416, ...
8.56 8 100 5 8 4*2+5*4+9*8   5 x 4.1 48 8, 4.590, 4.590, ...
8.57 8 100 5 20 5*20   145 x 3.1 48 8, 4.449, 4.427, ...
8.58 8 104 5 32 3*8+3*16+32   4 x 4.1
1 x 4.2		 48 8, 4.962, 4.835, ...
8.59 8 104 5 48 8+4*24   250 x 3.1 48 8, 4.768, 4.768, ...
8.60 8 104 5 96 4+12+16+24+48   12 x 4.1
9 x 4.2		 48 8, 4.962, 4.962, ...
8.61 8 104 5 1536 16+24+64   48 x 4.1
9 x 4.2		 44 8, 4.576, 4.576, ...
8.62 8 112 4 21504 tra   64 x 4.1
42 x 4.2		 50 8, 4, 4, ...
8.63 8 112 5 16 4*8+5*16 7.12 x K2 2 x 7.12 56 8, 6, 4.582, ...
8.64 8 112 5 16 4*8+5*16 7.15 x K2 2 x 7.15 56 8, 6, 4.534, ...
8.65 8 112 5 48 4+12+4*24 7.14 x K2 2 x 7.14 56 8, 6, 4.462, ...
8.66 8 112 5 32 3*16+2*32 7.17 x K2 2 x 7.17 56 8, 6, 4.494, ...
8.67 8 112 5 32 2*8+2*16+2*32 7.11 x K2 2 x 7.11 56 8, 6, 4.462, ...
8.68 8 112 5 32 2*8+2*16+2*32 7.13 x K2 2 x 7.13 56 8, 6, 4.606, ...
8.69 8 112 5 8 14*8 7.16 x K2 2 x 7.16 56 8, 6, 4.549, ...
8.70 8 116 6 672 4+28+84   84 x 4.1 56 8, 4.690, 4.690, ...
8.71 8 128 4 5160960 tra folded 28 56 x 6.13 56 8, 4, 4, ... (integral)
8.72 8 128 5 12288 tra 7.20 x K2 4 x 7.20 64 8, 6, 6, ... (integral)
8.73 8 128 5 6144 tra 7.21 x K2 4 x 7.21 64 8, 6, 6, ... (integral)
8.74 8 128 5 184320 tra 7.23 x K2 4 x 7.23 64 8, 6, 6, ... (integral)
8.75 8 128 5 92160 64+64 ebipd(N7.53) 2 x 6.3
31 x 6.13		 64 8, 6, 4, ... (integral)
8.76 8 128 5 96 32+96 7.18 x K2 2 x 7.18 64 8, 6, 4.711, ...
8.77 8 128 5 2304 tra 7.22 x K2 2 x 7.22 64 8, 6, 4.606, ...
8.78 8 128 5 240 48+80 7.19 x K2 2 x 7.19 64 8, 6, 4.692, ...
8.79 8 128 6 16 8*8+4*16 7.26 x K2 2 x 7.26 64 8, 6, 4.952, ...
8.80 8 128 6 48 8+5*24 7.25 x K2 2 x 7.25 64 8, 6, 4.988, ...
8.81 8 128 6 24 8+5*24 7.24 x K2 2 x 7.24 64 8, 6, 4.739, ...
8.82 8 128 6 24 2*4+2*12+4*24 7.27 x K2 2 x 7.27 64 8, 6, 4.828, ...
8.83 8 128 6 672 16+112 7.28 x K2 2 x 7.28 64 8, 6, 4.828, ...
8.84 8 128 6 384 16+2*24+64   64 x 4.1
32 x 4.2		 62 8, 4.899, 4.899, ...
8.85 8 128 6 48 2+6+8+16+2*48   64 x 4.1
8 x 4.2		 64 8, 4.788, 4.752, ...
8.86 8 128 6 64 2*8+16+32+64   64 x 4.1
16 x 4.2		 64 8, 4.899, 4.765, ...
8.87 8 128 6 512 2*16+32+64   64 x 4.1
24 x 4.2		 62 8, 4.899, 4.899, ...
8.88 8 128 6 10752 8+56+64   64 x 4.1
56 x 4.2		 58 8, 4.899, 4.899, ...
8.89 8 132 6 16 9*4+8*8+2*16   1 x 6.7 64 8, 5.212, 4.925, ...
8.90 8 132 6 72 2*12+36+72   6 x 5.2 64 8, 4.936, 4.936, ...
8.91 8 144 6 384 48+96 7.32 x K2 4 x 7.32 72 8, 6, 6, ...
8.92 8 144 6 768 48+96 7.30 x K2 4 x 7.30 72 8, 6, 6, ...
8.93 8 144 6 34560 tra 7.31 x K2 4 x 7.31 72 8, 6, 6, ...
8.94 8 144 6 24192 tra 7.29 x K2 2 x 7.29 72 8, 6, 4.606, ...
8.95 8 160 6 384 32+32+96 7.34 x K2 4 x 7.34 80 8, 6, 6, ...
8.96 8 160 6 960 tra 7.35 x K2 4 x 7.35 80 8, 6, 6, ...
8.97 8 160 6 96 16+48+96 7.33 x K2 2 x 7.33 80 8, 6, 5.236, ...
8.98 8 164 6 512 4+2*16+2*32+64   9 x 6.10 80 8, 5.475, 5.475, ...
8.99 8 176 6 63360 tra 7.36 x K2 6 x 7.36 88 8, 6, 6, ...
8.100 8 192 7 23040 tra 7.38 x K2 6 x 7.38 96 8, 6, 6, ...
8.101 8 192 7 2048 64+128 7.37 x K2 4 x 7.37 96 8, 6, 6, ...
8.102 8 196 6 225792 tra 4.1 x 4.1 42 x 6.12 96 8, 5.414, 5.414, ...
8.103 8 224 7 129024 tra 7.39 x K2 8 x 7.39 112 8, 6, 6, ...
8.104 8 256 8 10321920 tra 28 16 x 7.40 128 8, 6, 6, ... (integral)

Nonbipartite (0,2)-graphs of valency 0-8

In the table below, the bipd# column gives the serial number of the bipartite double of the graph.

#bipd #kv d|G|orbitsgraph subgraphs2-rkspectrum
N3.1 3.1 3 4 1 24 tra K4   1 3, -1, -1, -1
N4.1 4.2 4 8 2 48 tra N3.1 x K2 2 x N3.1 4 4, 2, ..., -2 (integral)
N5.1 5.2 5 12 3 120 tra icosahedron   6 5, 2.236, ..., -2.236
N5.2 5.4 5 16 2 1920 tra folded 25 20 x 3.1 6 5, 1, ..., -3 (integral)
N5.3 5.4 5 16 3 192 tra N4.1 x K2 4 x N4.1 8 5, 3, ..., -3 (integral)
N5.4 5.4 5 16 3 96 8+8   7 x 3.1
2 x N3.1		 8 5, 3, ..., -3 (integral)
N6.1 6.3 6 16 2 1152 tra K4 x K4 12 x N4.1 6 6, 2, ..., -2 (integral)
N6.2 6.3 6 16 2 192 tra Shrikhande   6 6, 2, ..., -2 (integral)
N6.3 6.8 6 20 3 60 tra   5 x N3.1 10 6, 2.449, ..., -3.236
N6.4 6.7 6 20 3 24 4+4+12   1 x N3.1 10 6, 3.020, ..., -3.020
N6.5 6.11 6 24 3 32 8+16   4 x 3.1
4 x N3.1		 12 6, 3.236, ..., -4
N6.6 6.11 6 24 3 480 tra folded 6.11 30 x 3.1 12 6, 2, ..., -4
N6.7 6.11 6 24 4 240 tra N5.1 x K2 2 x N5.1 12 6, 4, ..., -3.236
N6.8 6.12 6 28 3 48 4+12+12   2 x 4.1
3 x N4.1		 14 6, 3.414, ..., -4
N6.9 6.13 6 32 3 3840 tra N5.2 x K2 2 x N5.2 16 6, 4, ..., -4 (integral)
N6.10 6.13 6 32 4 1152 tra N5.3 x K2 6 x N5.3 16 6, 4, ..., -4 (integral)
N6.11 6.13 6 32 4 192 16+16 N5.4 x K2 2 x N5.4 16 6, 4, ..., -4 (integral)
N7.1 7.1 7 24 2 96 tra   2 x N5.1 12 7, 3, ..., -3
N7.2 7.1 7 24 2 96 tra   2 x N5.1 10 7, 3, ..., -3
N7.3 7.2 7 24 2 480 tra folded N7.46 2 x N5.1 12 7, 3, ..., -3
N7.4 7.2 7 24 2 480 tra   2 x N5.1 8 7, 3, ..., -3
N7.5 7.1 7 24 3 96 tra   6 x N4.1 12 7, 3, ..., -3
N7.6 7.2 7 24 3 48 tra   3 x 3.1 12 7, 3, ..., -3
N7.7 7.4 7 24 3 8 4*4+8   1 x 3.1 12 7, 3, ..., -3
N7.8 7.5 7 24 3 16 8+8+8   1 x 3.1 12 7, 3, ..., -3
N7.9 7.6 7 24 3 4 6*4     12 7, 2.909, ..., -3
N7.10 7.8 7 24 3 12 4*6     11 7, 3, ..., -3
N7.11 7.9 7 24 3 12 4*6   3 x 3.1 10 7, 2.798, ..., -3
N7.12 7.10 7 24 3 336 tra drg(7,4,1; 1,2,7)   9 7, 2.646, ..., -2.646
N7.13 7.24 7 32 3 6 2+5*6   3 x 3.1
3 x N3.1		 14 7, 3.661, ..., -3.739
N7.14 7.26 7 32 3 4 8*2+4*4   1 x 3.1
3 x N3.1		 15 7, 3.839, ..., -3.952
N7.15 7.27 7 32 3 2 8*1+12*2   3 x N3.1 16 7, 3.638, ..., -3.828
N7.16 7.18 7 32 3 8 4*8     16 7, 3.711, ..., -3.711
N7.17 7.18 7 32 3 24 8+24   8 x N3.1 14 7, 3.606, ..., -3.711
N7.18 7.20 7 32 3 512 tra   2 x N5.2
4 x N5.3		 16 7, 3, ..., -5 (integral)
N7.19 7.20 7 32 3 256 tra   2 x N5.3
4 x N5.4		 16 7, 3, ..., -5 (integral)
N7.20 7.21 7 32 3 96 tra   2 x 4.2
4 x N4.1		 16 7, 3, ..., -5 (integral)
N7.21 7.21 7 32 3 768 tra   2 x N5.2 14 7, 3, ..., -5 (integral)
N7.22 7.28 7 32 3 24 4+4+24   8 x N3.1 16 7, 3.494, ..., -3.828
N7.23 7.28 7 32 3 168 4+28     16 7, 3.494, ..., -3.828
N7.24 7.22 7 32 3 48 8+24     14 7, 3.606, ..., -3.606
N7.25 7.23 7 32 3 1536 tra   2 x N5.2
12 x N5.3		 16 7, 3, ..., -5 (integral)
N7.26 7.23 7 32 3 256 tra   2 x N5.3
4 x N5.4		 16 7, 3, ..., -5 (integral)
N7.27 7.23 7 32 3 2304 tra N6.1 x K2 2 x N6.1 16 7, 5, ..., -3 (integral)
N7.28 7.23 7 32 3 384 tra N6.2 x K2 2 x N6.2 16 7, 5, ..., -3 (integral)
N7.29 7.25 7 32 4 12 2+5*6   3 x 3.1
2 x N3.1		 15 7, 3.988, ..., -3.686
N7.30 7.28 7 32 4 56 4+28     16 7, 3.828, ..., -3.828
N7.31 7.30 7 36 3 32 4+2*8+16   25 x 3.1
5 x N3.1		 17 7, 3.449, ..., -5
N7.32 7.32 7 36 3 8 3*4+3*8   25 x 3.1
5 x N3.1		 17 7, 3.732, ..., -5
N7.33 7.34 7 40 4 48 8+8+24 N6.4 x K2 2 x N6.4 20 7, 5, ..., -4.020
N7.34 7.34 7 40 4 16 3*8+16   1 x N4.1 18 7, 4.020, ..., -5
N7.35 7.35 7 40 4 8 5*8   1 x N4.1 20 7, 4.236, ..., -5
N7.36 7.35 7 40 4 120 tra N6.3 x K2 2 x N6.3 20 7, 5, ..., -4.236
N7.37 7.36 7 44 3 80 4+20+20   2 x 5.1 21 7, 3.732, ..., -5
N7.38 7.37 7 48 4 32 16+16+16   8 x 4.1 24 7, 4.236, ..., -5
N7.39 7.37 7 48 4 32 4*8+16   120 x 3.1
4 x N3.1		 24 7, 4.464, ..., -5
N7.40 7.37 7 48 4 128 16+32   120 x 3.1
4 x N3.1		 22 7, 4.464, ..., -5
N7.41 7.38 7 48 4 64 16+32 N6.5 x K2 2 x N6.5 24 7, 5, ..., -5
N7.42 7.38 7 48 4 96 24+24   2 x 5.2 22 7, 4.236, ..., -5
N7.43 7.38 7 48 4 64 16+16+16   2 x 5.2 24 7, 4.236, ..., -5
N7.44 7.38 7 48 4 960 tra   2 x 5.2 24 7, 4.236, ..., -5
N7.45 7.38 7 48 4 960 tra N6.6 x K2 2 x N6.6 24 7, 5, ..., -5
N7.46 7.38 7 48 5 960 tra N6.7 x K2 4 x N6.7 24 7, 5, ..., -4.236
N7.47 7.38 7 48 5 480 24+24   1 x 5.2
2 x N5.1		 24 7, 5, ..., -5
N7.48 7.39 7 56 4 96 8+24+24 N6.8 x K2 2 x N6.8 28 7, 5, ..., -5
N7.49 7.39 7 56 4 8064 tra 4.1 x K4 6 x 5.3
21 x N5.3		 26 7, 4.414, ..., -5
N7.50 7.39 7 56 4 384 8+16+32   6 x 5.3
1 x N5.3
4 x N5.4		 26 7, 4.414, ..., -5
N7.51 7.40 7 64 3 322560 tra folded 27 42 x 5.4 28 7, 3, ..., -5 (integral)
N7.52 7.40 7 64 4 15360 tra N6.9 x K2 4 x N6.9 32 7, 5, ..., -5 (integral)
N7.53 7.40 7 64 4 7680 32+32   21 x 5.4
2 x N5.2		 32 7, 5, ..., -5 (integral)
N7.54 7.40 7 64 5 9216 tra N6.10 x K2 8 x N6.10 32 7, 5, ..., -5 (integral)
N7.55 7.40 7 64 5 768 32+32 N6.11 x K2 4 x N6.11 32 7, 5, ..., -5 (integral)
N7.56 7.40 7 64 5 768 16+16+32   8 x 5.4
4 x N5.4		 32 7, 5, ..., -5 (integral)
N8.1 8.3 8 34 3 2 17*2   1 x 3.1 16 8, 3.514, ..., -3.708
N8.2 8.3 8 34 3 2 17*2   1 x 3.1 16 8, 3.611, ..., -3.708
N8.3 8.4 8 34 3 10 2*2+3*10   1 x N3.1 16 8, 3.393, ..., -3.689
N8.4 8.7 8 36 3 72 tra   36 x 3.1
18 x N3.1		 16 8, 3.646, ..., -4
N8.5 8.7 8 36 3 12 12+12+12   3 x 3.1 16 8, 4, ..., -4
N8.6 8.7 8 36 3 12 4*6+12   3 x 3.1 16 8, 4, ..., -4
N8.7 8.8 8 37 3 12 2*1+2+3+3*6+12   5 x 3.1 14 8, 3.646, ..., -3.646
N8.8 8.8 8 37 3 336 1+8+28   56 x 3.1
14 x N3.1		 14 8, 3.414, ..., -3.646
N8.9 8.13 8 40 3 1920 tra folded 8.13 40 x 3.1 16 8, 2, ..., -4 (integral)
N8.10 8.13 8 40 3 384 8+32   32 x 3.1
16 x N3.1		 16 8, 4, ..., -4 (integral)
N8.11 8.11 8 40 3 10 4*10   15 x 3.1
15 x N3.1		 18 8, 4.179, ..., -4.512
N8.12 8.12 8 40 3 32 8+16+16     18 8, 4, ..., -4
N8.13 8.12 8 40 3 96 16+24   8 x 3.1
16 x N3.1		 18 8, 4, ..., -4
N8.14 8.15 8 40 3 192 8+32   4 x 3.1
8 x N3.1		 18 8, 3.464, ..., -4
N8.15 8.13 8 40 4 192 16+24     16 8, 4, ..., -4 (integral)
N8.16 8.13 8 40 4 32 3*8+16   12 x 3.1 16 8, 4, ..., -4 (integral)
N8.17 8.15 8 40 4 64 8+32     18 8, 4, ..., -4
N8.18 8.24 8 42 3 8 2+6*4+2*8   16 x 3.1
8 x N3.1		 20 8, 3.805, ..., -4.256
N8.19 8.25 8 42 3 8 3*2+3*4+3*8   6 x 3.1 20 8, 4.218, ..., -4.337
N8.20 8.25 8 42 3 8 3*2+3*4+3*8   16 x 3.1
8 x N3.1		 20 8, 4.039, ..., -4.337
N8.21 8.29 8 42 3 48 6+12+24   12 x 3.1 20 8, 4.218, ..., -4.218
N8.22 8.29 8 42 3 16 2+2*4+2*8+16   20 x 3.1
8 x N3.1		 20 8, 4.218, ..., -4.218
N8.23 8.24 8 42 4 8 2+6*4+2*8   6 x 3.1 20 8, 4.256, ..., -4.110
N8.24 8.25 8 42 4 8 3*2+3*4+3*8     20 8, 4.313, ..., -4.337
N8.25 8.25 8 42 4 8 3*2+3*4+3*8   4 x 3.1
4 x N3.1		 20 8, 4.313, ..., -4.337
N8.26 8.26 8 42 4 8 3*2+3*4+3*8   8 x 3.1
4 x N3.1		 20 8, 4.337, ..., -4.337
N8.27 8.26 8 42 4 32 2+8+2*16   8 x 3.1 20 8, 4.307, ..., -4.337
N8.28 8.26 8 42 4 8 3*2+3*4+3*8   10 x 3.1 20 8, 4.337, ..., -4.337
N8.29 8.26 8 42 4 32 2+8+2*16   32 x 3.1
8 x N3.1		 20 8, 4.307, ..., -4.337
N8.30 8.29 8 42 4 16 2+2*4+2*8+16   4 x 3.1 20 8, 4.218, ..., -4.218
N8.31 8.29 8 42 4 48 6+12+24     20 8, 4.218, ..., -4.218
N8.32 8.31 8 44 3 4 6*2+8*4   1 x N4.1 20 8, 3.948, ..., -4.812
N8.33 8.32 8 44 3 4 6*2+8*4   1 x N4.1 20 8, 3.871, ..., -4.946
N8.34 8.33 8 48 3 48 24+24   2 x N5.1 24 8, 4, ..., -4
N8.35 8.34 8 48 3 2880 tra folded N8.186 6 x N6.7 24 8, 4, ..., -5.236
N8.36 8.34 8 48 3 480 24+24   1 x N6.6
1 x N6.7		 24 8, 4, ..., -5.236
N8.37 8.34 8 48 3 960 tra   2 x N6.6 24 8, 4, ..., -5.236
N8.38 8.37 8 48 3 192 tra N7.1 x K2 2 x N7.1 24 8, 6, ..., -4
N8.39 8.37 8 48 3 192 tra N7.2 x K2 2 x N7.2 24 8, 6, ..., -4
N8.40 8.37 8 48 3 32 16+32   2 x 5.2
2 x N5.3		 24 8, 4, ..., -6
N8.41 8.37 8 48 3 64 16+32   2 x 5.2 24 8, 4, ..., -6
N8.42 8.37 8 48 3 192 tra   2 x 5.2 24 8, 4, ..., -6
N8.43 8.38 8 48 3 960 tra N7.4 x K2 2 x N7.4 24 8, 6, ..., -4
N8.44 8.38 8 48 3 960 tra N7.3 x K2 2 x N7.3 24 8, 6, ..., -4
N8.45 8.38 8 48 3 96 tra   2 x 5.2
6 x N5.3		 24 8, 4, ..., -6
N8.46 8.38 8 48 3 64 16+32   2 x 5.2 24 8, 4, ..., -6
N8.47 8.38 8 48 3 960 tra   2 x 5.2 24 8, 3.236, ..., -6
N8.48 8.39 8 48 3 32 16+32   2 x N5.3 24 8, 4, ..., -6
N8.49 8.39 8 48 3 96 tra   6 x N5.3 24 8, 4, ..., -6
N8.50 8.40 8 48 3 16 2*4+8+2*16   2 x 4.2
2 x N4.1		 24 8, 4, ..., -6
N8.51 8.40 8 48 3 32 16+32   2 x N5.3 24 8, 4, ..., -6
N8.52 8.41 8 48 3 16 2*4+8+2*16   3 x 4.2 24 8, 3.798, ..., -6
N8.53 8.41 8 48 3 32 16+32   2 x N5.3 24 8, 4, ..., -6
N8.54 8.42 8 48 3 48 tra   18 x N4.1 24 8, 4, ..., -6
N8.55 8.42 8 48 3 48 tra   12 x N4.1 24 8, 4, ..., -6
N8.56 8.42 8 48 3 48 tra   6 x N4.1 24 8, 4, ..., -6
N8.57 8.43 8 48 3 24 12+12+24   12 x N4.1 24 8, 4, ..., -6
N8.58 8.44 8 48 3 96 tra   12 x N4.1 24 8, 3.646, ..., -6
N8.59 8.45 8 48 3 48 tra   18 x N4.1 24 8, 4, ..., -6
N8.60 8.50 8 48 3 48 24+24   60 x 3.1
6 x N3.1		 22 8, 4.243, ..., -4.429
N8.61 8.52 8 48 3 96 tra   12 x N4.1 24 8, 3.646, ..., -6
N8.62 8.34 8 48 4 2880 tra N5.1 x K4 6 x N6.7 24 8, 5.236, ..., -3.236
N8.63 8.34 8 48 4 64 16+16+16   2 x N6.5 24 8, 5.236, ..., -5.236
N8.64 8.36 8 48 4 4 6*2+9*4   10 x 3.1
8 x N3.1		 24 8, 4.417, ..., -4.494
N8.65 8.36 8 48 4 12 2*6+3*12   12 x 3.1 24 8, 4.494, ..., -4.340
N8.66 8.37 8 48 4 384 tra   2 x N6.6 24 8, 4, ..., -6
N8.67 8.37 8 48 4 128 16+32   2 x N6.5 24 8, 4, ..., -6
N8.68 8.37 8 48 4 192 tra N7.5 x K2 2 x N7.5 24 8, 6, ..., -4
N8.69 8.38 8 48 4 1920 tra   2 x N6.6 24 8, 4, ..., -6
N8.70 8.38 8 48 4 128 16+32   2 x N6.5 24 8, 4, ..., -6
N8.71 8.38 8 48 4 96 tra N7.6 x K2 2 x N7.6 24 8, 6, ..., -4
N8.72 8.39 8 48 4 24 4*12 N7.11 x K2 2 x N7.11 24 8, 6, ..., -4
N8.73 8.40 8 48 4 16 4*8+16 N7.7 x K2 2 x N7.7 24 8, 6, ..., -4
N8.74 8.41 8 48 4 32 16+16+16 N7.8 x K2 2 x N7.8 24 8, 6, ..., -4
N8.75 8.42 8 48 4 8 6*8 N7.9 x K2 2 x N7.9 24 8, 6, ..., -4
N8.76 8.43 8 48 4 24 4*12 N7.10 x K2 2 x N7.10 24 8, 6, ..., -4
N8.77 8.50 8 48 4 16 4*8+16   13 x 3.1 22 8, 4.429, ..., -4.243
N8.78 8.50 8 48 4 24 12+12+24   21 x 3.1 22 8, 4.429, ..., -4.429
N8.79 8.51 8 48 4 12 2*6+3*12     24 8, 4.415, ..., -4.415
N8.80 8.51 8 48 4 12 6*6+12   6 x 3.1 24 8, 4.415, ..., -4.415
N8.81 8.52 8 48 4 672 tra N7.12 x K2 2 x N7.12 24 8, 6, ..., -3.646
N8.82 8.58 8 52 4 8 3*4+5*8   2 x 4.1
1 x N4.1		 24 8, 4.780, ..., -4.962
N8.83 8.58 8 52 4 16 3*4+3*8+16   2 x 4.1
1 x N4.1		 24 8, 4.835, ..., -4.962
N8.84 8.58 8 52 4 16 3*4+3*8+16   2 x 4.1
1 x N4.1		 24 8, 4.835, ..., -4.962
N8.85 8.59 8 52 4 12 2*2+8*6   52 x 3.1
2 x N3.1		 24 8, 4.768, ..., -4.768
N8.86 8.59 8 52 4 8 5*4+4*8   42 x 3.1
10 x N3.1		 24 8, 4.309, ..., -4.768
N8.87 8.60 8 52 4 48 2+6+8+12+24   6 x 4.1
1 x N4.1		 24 8, 4.783, ..., -4.962
N8.88 8.60 8 52 4 16 2*2+2*4+3*8+16   6 x 4.1
1 x N4.1		 24 8, 4.783, ..., -4.962
N8.89 8.61 8 52 4 48 2+3*6+8+24   6 x 4.1
1 x N4.1		 22 8, 4.576, ..., -4.576
N8.90 8.63 8 56 4 8 4*4+5*8   1 x N4.1 28 8, 4.557, ..., -6
N8.91 8.63 8 56 4 8 4*4+5*8   1 x N4.1 28 8, 4.462, ..., -6
N8.92 8.66 8 56 4 16 3*8+2*16   88 x 3.1
6 x N3.1		 28 8, 4.482, ..., -6
N8.93 8.66 8 56 4 8 7*8   88 x 3.1
6 x N3.1		 28 8, 4.494, ..., -6
N8.94 8.67 8 56 4 16 2*4+2*8+2*16   1 x N4.1 28 8, 4.462, ..., -6
N8.95 8.68 8 56 4 16 2*4+2*8+2*16   1 x N4.1 28 8, 4.462, ..., -6
N8.96 8.69 8 56 4 4 14*4   1 x N4.1 28 8, 4.356, ..., -6
N8.97 8.71 8 64 3 46080 tra N5.2 x K4 6 x N6.9
20 x N6.10		 28 8, 4, ..., -4 (integral)
N8.98 8.71 8 64 3 4608 tra   2 x N6.10
12 x N6.11		 28 8, 4, ..., -4 (integral)
N8.99 8.72 8 64 3 256 16+16+32   2 x 6.1
2 x N6.10
2 x N6.11		 32 8, 4, ..., -6 (integral)
N8.100 8.72 8 64 3 128 16+16+32   2 x 6.1 32 8, 4, ..., -6 (integral)
N8.101 8.72 8 64 3 256 16+16+32   2 x 6.1
2 x N6.9
2 x N6.11		 32 8, 4, ..., -6 (integral)
N8.102 8.73 8 64 3 128 32+32   2 x 6.2 32 8, 4, ..., -6 (integral)
N8.103 8.73 8 64 3 64 4*16   2 x 6.2
2 x N6.11		 32 8, 4, ..., -6 (integral)
N8.104 8.74 8 64 3 512 32+32   2 x 6.3
4 x N6.11		 32 8, 4, ..., -6 (integral)
N8.105 8.74 8 64 3 15360 tra   2 x 6.3
20 x N6.10		 32 8, 4, ..., -6 (integral)
N8.106 8.74 8 64 3 768 32+32   2 x 6.3
1 x N6.9
1 x N6.10
6 x N6.11		 32 8, 4, ..., -6 (integral)
N8.107 8.74 8 64 3 1536 tra   2 x 6.3 32 8, 4, ..., -6 (integral)
N8.108 8.74 8 64 3 9216 tra   2 x 6.3
12 x N6.9		 32 8, 4, ..., -6 (integral)
N8.109 8.75 8 64 3 7680 32+32   1 x 6.3
1 x N6.9
20 x N6.10		 32 8, 4, ..., -6 (integral)
N8.110 8.75 8 64 3 384 4*16   1 x 6.3
1 x N6.10
8 x N6.11		 32 8, 4, ..., -6 (integral)
N8.111 8.75 8 64 3 2304 32+32   1 x 6.3
6 x N6.9
1 x N6.10
6 x N6.11		 32 8, 4, ..., -6 (integral)
N8.112 8.77 8 64 3 1152 tra   8 x N5.2 32 8, 4, ..., -6
N8.113 8.72 8 64 4 1024 tra N7.18 x K2 2 x N7.18 32 8, 6, ..., -6 (integral)
N8.114 8.72 8 64 4 512 tra N7.19 x K2 2 x N7.19 32 8, 6, ..., -6 (integral)
N8.115 8.73 8 64 4 1536 tra N7.21 x K2 2 x N7.21 32 8, 6, ..., -6 (integral)
N8.116 8.73 8 64 4 192 tra N7.20 x K2 2 x N7.20 32 8, 6, ..., -6 (integral)
N8.117 8.74 8 64 4 1536 tra N7.28 x K2 4 x N7.28 32 8, 6, ..., -4 (integral)
N8.118 8.74 8 64 4 9216 tra N7.27 x K2 4 x N7.27 32 8, 6, ..., -4 (integral)
N8.119 8.74 8 64 4 768 32+32   1 x 6.3
2 x N6.2		 32 8, 6, ..., -6 (integral)
N8.120 8.74 8 64 4 4608 32+32   1 x 6.3
2 x N6.1
12 x N6.11		 32 8, 6, ..., -6 (integral)
N8.121 8.74 8 64 4 3072 tra N7.25 x K2 2 x N7.25 32 8, 6, ..., -6 (integral)
N8.122 8.74 8 64 4 512 tra N7.26 x K2 2 x N7.26 32 8, 6, ..., -6 (integral)
N8.123 8.75 8 64 4 2304 32+32 N5.4 x K4 2 x N6.1
7 x N6.10
6 x N6.11		 32 8, 6, ..., -4 (integral)
N8.124 8.76 8 64 4 16 4*16 N7.16 x K2 2 x N7.16 32 8, 6, ..., -4.711
N8.125 8.76 8 64 4 16 4*16   124 x 3.1
8 x N3.1		 32 8, 4.711, ..., -6
N8.126 8.76 8 64 4 48 16+48 N7.17 x K2 2 x N7.17 32 8, 6, ..., -4.711
N8.127 8.76 8 64 4 16 4*16   140 x 3.1
8 x N3.1		 32 8, 4.711, ..., -6
N8.128 8.76 8 64 4 48 16+48   224 x 3.1 32 8, 4.494, ..., -6
N8.129 8.77 8 64 4 96 16+48 N7.24 x K2 2 x N7.24 32 8, 6, ..., -4.606
N8.130 8.77 8 64 4 64 32+32   8 x 4.2 32 8, 4.606, ..., -6
N8.131 8.77 8 64 4 96 16+48   232 x 3.1 32 8, 4.606, ..., -6
N8.132 8.78 8 64 4 8 8*8   3 x N4.1 32 8, 4.692, ..., -6
N8.133 8.79 8 64 4 8 8*4+4*8 N7.14 x K2 2 x N7.14 32 8, 6, ..., -4.952
N8.134 8.79 8 64 4 8 8*4+4*8   1 x 4.2
1 x N4.1		 32 8, 4.839, ..., -6
N8.135 8.80 8 64 4 8 8*4+4*8   1 x 4.2 32 8, 4.988, ..., -6
N8.136 8.81 8 64 4 4 16*4   2 x 4.1
1 x 4.2
1 x N4.1		 32 8, 4.684, ..., -6
N8.137 8.81 8 64 4 12 4+5*12 N7.13 x K2 2 x N7.13 32 8, 6, ..., -4.739
N8.138 8.82 8 64 4 4 8*2+12*4 N7.15 x K2 2 x N7.15 32 8, 6, ..., -4.828
N8.139 8.83 8 64 4 16 2*8+3*16   124 x 3.1
8 x N3.1		 32 8, 4.828, ..., -6
N8.140 8.83 8 64 4 112 8+56   224 x 3.1 32 8, 4.828, ..., -6
N8.141 8.83 8 64 4 336 8+56 N7.23 x K2 2 x N7.23 32 8, 6, ..., -4.828
N8.142 8.83 8 64 4 48 8+8+48 N7.22 x K2 2 x N7.22 32 8, 6, ..., -4.828
N8.143 8.80 8 64 5 24 4+5*12 N7.29 x K2 2 x N7.29 32 8, 6, ..., -4.686
N8.144 8.83 8 64 5 112 8+56 N7.30 x K2 2 x N7.30 32 8, 6, ..., -4.828
N8.145 8.91 8 72 4 16 3*8+3*16 N7.32 x K2 2 x N7.32 36 8, 6, ..., -6
N8.146 8.91 8 72 4 32 2*4+4*8+2*16   2 x 6.6 36 8, 4.828, ..., -6
N8.147 8.91 8 72 4 32 3*8+3*16   2 x 6.6 36 8, 4.828, ..., -6
N8.148 8.91 8 72 4 192 24+48   2 x 6.6 36 8, 4.828, ..., -6
N8.149 8.91 8 72 4 192 24+48   2 x 6.6 36 8, 4.828, ..., -6
N8.150 8.92 8 72 4 64 8+2*16+32 N7.31 x K2 2 x N7.31 36 8, 6, ..., -6
N8.151 8.92 8 72 4 32 5*8+2*16   2 x 6.4 36 8, 4.828, ..., -6
N8.152 8.92 8 72 4 96 24+24+24   2 x 6.4 36 8, 4.828, ..., -6
N8.153 8.93 8 72 4 160 4+8+20+40   2 x 6.5 36 8, 4.449, ..., -6
N8.154 8.95 8 80 4 32 6*8+2*16   2 x 6.7 40 8, 5.020, ..., -6
N8.155 8.95 8 80 4 192 16+16+48   2 x 6.7 40 8, 5.020, ..., -6
N8.156 8.95 8 80 4 64 3*16+32   2 x 6.7 40 8, 5.020, ..., -6
N8.157 8.96 8 80 4 32 5*16   2 x 6.8 40 8, 5.236, ..., -6
N8.158 8.96 8 80 4 480 tra   2 x 6.8 40 8, 4.449, ..., -6
N8.159 8.97 8 80 4 16 4*8+3*16   1 x N6.5 40 8, 5.236, ..., -6
N8.160 8.97 8 80 4 16 4*8+3*16   1 x N6.5 40 8, 5.236, ..., -6
N8.161 8.95 8 80 5 192 16+16+48 N7.33 x K2 4 x N7.33 40 8, 6, ..., -5.020
N8.162 8.95 8 80 5 96 4*8+2*24   1 x 6.7
2 x N6.4		 40 8, 6, ..., -6
N8.163 8.95 8 80 5 32 3*16+32 N7.34 x K2 2 x N7.34 40 8, 6, ..., -6
N8.164 8.96 8 80 5 16 5*16 N7.35 x K2 2 x N7.35 40 8, 6, ..., -6
N8.165 8.96 8 80 5 480 tra N7.36 x K2 4 x N7.36 40 8, 6, ..., -5.236
N8.166 8.96 8 80 5 240 40+40   1 x 6.8
2 x N6.3		 40 8, 6, ..., -6
N8.167 8.99 8 88 4 160 8+40+40 N7.37 x K2 2 x N7.37 44 8, 6, ..., -6
N8.168 8.99 8 88 4 31680 tra 5.1 x K4 6 x 6.9 44 8, 4.732, ..., -6
N8.169 8.99 8 88 4 576 16+24+48   6 x 6.9 44 8, 4.732, ..., -6
N8.170 8.100 8 96 4 768 32+64   6 x 6.11 48 8, 5.236, ..., -6
N8.171 8.100 8 96 4 11520 tra folded 8.100 6 x 6.11 48 8, 4, ..., -6
N8.172 8.100 8 96 5 256 32+64 N7.41 x K2 4 x N7.41 48 8, 6, ..., -6
N8.173 8.100 8 96 5 128 2*16+2*32   1 x 6.11
2 x N6.5		 48 8, 6, ..., -6
N8.174 8.100 8 96 5 192 48+48 N7.42 x K2 2 x N7.42 48 8, 6, ..., -6
N8.175 8.100 8 96 5 128 32+32+32 N7.43 x K2 2 x N7.43 48 8, 6, ..., -6
N8.176 8.100 8 96 5 11520 tra 5.2 x K4 6 x 6.11 48 8, 5.236, ..., -6
N8.177 8.100 8 96 5 768 32+64   6 x 6.11 48 8, 5.236, ..., -6
N8.178 8.100 8 96 5 3840 tra N7.45 x K2 4 x N7.45 48 8, 6, ..., -6
N8.179 8.100 8 96 5 1920 48+48   1 x 6.11
2 x N6.6		 48 8, 6, ..., -6
N8.180 8.100 8 96 5 1920 tra N7.44 x K2 2 x N7.44 48 8, 6, ..., -6
N8.181 8.101 8 96 5 128 2*8+16+2*32   2 x 6.10
4 x N6.8		 48 8, 5.464, ..., -6
N8.182 8.101 8 96 5 64 4*8+2*16+32   2 x 6.10
2 x N6.8		 48 8, 5.464, ..., -6
N8.183 8.101 8 96 5 64 32+32+32 N7.38 x K2 2 x N7.38 48 8, 6, ..., -6
N8.184 8.101 8 96 5 64 4*16+32 N7.39 x K2 2 x N7.39 48 8, 6, ..., -6
N8.185 8.101 8 96 5 256 32+64 N7.40 x K2 2 x N7.40 48 8, 6, ..., -6
N8.186 8.100 8 96 6 5760 tra N7.46 x K2 6 x N7.46 48 8, 6, ..., -5.236
N8.187 8.100 8 96 6 960 48+48 N7.47 x K2 2 x N7.47 48 8, 6, ..., -6
N8.188 8.103 8 112 4 2304 16+48+48   12 x 6.12
3 x N6.9		 56 8, 5.414, ..., -6
N8.189 8.103 8 112 5 384 16+48+48 N7.48 x K2 4 x N7.48 56 8, 6, ..., -6
N8.190 8.103 8 112 5 192 2*8+4*24   3 x 6.12
2 x N6.8
3 x N6.11		 56 8, 6, ..., -6
N8.191 8.103 8 112 5 16128 tra N7.49 x K2 2 x N7.49 56 8, 6, ..., -6
N8.192 8.103 8 112 5 768 16+32+64 N7.50 x K2 2 x N7.50 56 8, 6, ..., -6
N8.193 8.104 8 128 4 645120 tra N7.51 x K2 2 x N7.51 64 8, 6, ..., -6 (integral)
N8.194 8.104 8 128 5 92160 tra N7.52 x K2 6 x N7.52 64 8, 6, ..., -6 (integral)
N8.195 8.104 8 128 5 15360 64+64 N7.53 x K2 2 x N7.53 64 8, 6, ..., -6 (integral)
N8.196 8.104 8 128 6 92160 tra N7.54 x K2 10 x N7.54 64 8, 6, ..., -6 (integral)
N8.197 8.104 8 128 6 4608 64+64 N7.55 x K2 6 x N7.55 64 8, 6, ..., -6 (integral)
N8.198 8.104 8 128 6 1536 32+32+64 N7.56 x K2 2 x N7.56 64 8, 6, ..., -6 (integral)

References

A. E. Brouwer, Classification of small (0,2)-graphs, J. Combinatorial Theory (A) 113 (2006) 1636-1645. DVI

A. E. Brouwer & P. R. J. Östergård, Classification of the (0,2)-graphs of valency 8, preprint. DVI