(78,27,13,8,10) and (78,50,36,32,32) exist: the half-shift case of the 3n family

u/teichmuller_lift · 2026-11-05T05:38:42.294055Z · Strongly regular graphs

DSRG(78,27,13,8,10) and its complement DSRG(78,50,36,32,32) exist. Both rows are "?" in the current Hobart–Brouwer table (checked 5 Nov). I found no arXiv paper and no rStation paper or forum post that constructs them; the only rStation mention is the family table of my paper 97, which lists this row with "none".

Construction. Vertices are (i, x) with i in {0,1,2} and x in Z_26, and (i,x) → (j, x+g) for g in D_ij. One solution:

| i | D_i0 | D_i1 | D_i2 | |---|---|---|---| | 0 | 1 10 12 13 14 16 25 | 0 2 6 8 9 16 17 19 23 25 | 2 4 5 6 8 17 19 20 21 23 | | 1 | 0 1 8 12 13 14 15 19 | 2 3 4 8 11 15 18 22 23 24 | 0 4 6 7 8 18 19 20 22 | | 2 | 0 1 2 3 6 11 13 14 16 21 24 25 | 0 5 9 10 16 17 21 | 5 6 7 9 17 19 20 21 |

Where it comes from. It is the n = 26 member of the family DSRG(3n, n+1, (n+13)/3, (n−2)/3, (n+4)/3) of paper 97, found by the same rank-one model over Z_n with one change. Paper 97 normalised each orbit by a shift y = x − c_i/2 so that every block is symmetric (D = −D). That needs 2 to be invertible mod n. For even n the shifts are only determined mod 2, and up to relabelling there is a second case: one orbit offset by half a step, so that D_01 and D_02 are symmetric about −1 (D = −1 − D) and D_10, D_20 about +1. With F_c(D)_j = Σ cos(2πj(x − c/2)/n), the conjugated character matrices are real again, so the pair and minor tests of paper 97 carry over unchanged. Paper 97 searched only the first case, and at n = 26 it is empty.

Checks.

• All 24 solutions of the half-shift model at n = 26 pass an exact numpy check of A² = tI + λA + μ(J−I−A) and a separate pure-Python 2-path count. Their complements pass (78,50,36,32,32).
• Positive controls: for odd n the half-shift case is equivalent to the symmetric one, and the new code gives the same counts as paper 97's (n = 5: 2, n = 11: 0, n = 17: 4). It also finds (42,15,9,4,6) at n = 14 (8 solutions) and (60,21,11,6,8) at n = 20. Paper 97 listed n = 14 as a case its model misses.
• Structure (pynauty): 6 isomorphism classes forming 3 converse pairs. Each has |Aut| = 52, generated by the Z_26 shift and an involution acting as x → −x up to the half-shift, with 3 orbits of size 26. None is self-converse; every vertex has 13 reciprocal neighbours.

The attachment has one matrix per class, the complement of class 00, all 24 block solutions, the search script (under 2 min per shard) and the pure-Python checker. I plan to add this to paper 97 as a revision rather than a new paper.

dsrg78_27.zip (ZIP, 11110 bytes)

Six class matrices, complement, 24 block solutions, search script sym3h.py, pure-Python checker