Three new 4-regular graphs of diameter 3 on 41 vertices, with symmetry analysis of the known (4,3) graphs of order 41

The largest currently known 4-regular graphs of diameter 3 have 41 vertices (J. Allwright, Discrete Appl. Math. 37/38 (1992) 3–8, who found 13 non-isomorphic such graphs by heuristic search). This record presents three further pairwise non-isomorphic (4,3) graphs on 41 vertices, N1, N2 and N3, together with a symmetry analysis of the 16 known graphs. We are unaware of any previously published 41-vertex examples outside Allwright's 13.

Highlights. The 16 graphs fall into two families: ten with exactly one triangle (Allwright's 1(a)–1(f), 2(a)–2(c), and N3) and six triangle-free graphs of girth 5 (Allwright's graphs 3–6, N1 and N2); the latter have average distance exactly 2.5, the minimum possible for a 4-regular graph of diameter 3 on 41 vertices. In the known graphs, the fixed vertices of an involution typically form the shortest cycle (the triangle, or a 5-cycle). Enumerating all graphs with such an involution shows that N3 and N2 complete classes otherwise populated by Allwright's graphs (8 of 9 and 3 of 4). The closed families also explain the graphs without these symmetries: 1(d) lies between 1(c) and 1(e) on a mirror-symmetric pair of Allwright's own 2-swaps, and within the triangle-free family N1 and graph 3 differ from N2 only in two and one of three "switch" positions respectively.

Graphs. N1: triangle-free, girth 5, |Aut| = 3 (C3). N2: triangle-free, girth 5, |Aut| = 6 (S3). N3: exactly one triangle, |Aut| = 6 (C6; its involution and order-3 automorphism commute). Files are 0-based edge lists. Allwright's 13 graphs are available as adjacency lists on F. Comellas's degree–diameter table page; all comparisons below use those files (they agree with Allwright's published constructions: his stated 2-swaps, applied to 1(a) and 2(a), reproduce 1(b)–1(f) and 2(b)–2(c)).

Verification. All 16: 41 vertices, 82 edges, simple, connected, 4-regular, diameter 3 (breadth-first search; for N1–N3 also F. Comellas's NetworkX checker, average distance 2.500000, 2.500000, 2.503659). Pairwise non-isomorphic (16 distinct nauty canonical forms; also 16 distinct vectors of k-cycle counts for k = 3..10 — note that N1 and N2 agree for k ≤ 8 and first differ at k = 9). |Aut| by nauty and an independent backtracking count. Validity, non-isomorphism, |Aut| and cycle counts were also reproduced by separately written code (produced with a second AI assistant).

Symmetry classes (computer enumeration). Each statement below comes from a complete SAT enumeration of all labelled graphs with a prescribed automorphism, using lexicographic symmetry breaking; every solution was re-checked by BFS and classified by nauty. Every (4,3) graph on 41 vertices that has
 (a) an involution whose fixed vertices are exactly the vertices of a triangle is one of 9 graphs: Allwright's 1(a), 1(b), 1(c), 1(e), 1(f), 2(a), 2(b), 2(c), and N3;
 (b) no triangle and an involution whose fixed vertices are exactly a 5-cycle is one of 4: Allwright's graphs 4, 5 and 6, and N2;
 (c) no triangle and an involution whose fixed vertices are exactly a 7-cycle is Allwright's graph 6;
 (d) an automorphism of order 3 fixing exactly two vertices is one of 6: Allwright's 2(b), 2(c) and graph 6, together with N1, N2 and N3 — obtained by two different enumerators (a direct one, and a quotient-level enumerator validated against geng and a naive SAT enumeration for n = 11–26).
Statements (a)–(c) rest on one encoding; an independent check had not been completed at the time of this deposit. These classes cover the main symmetry types of the known graphs; other involution fixed-point shapes occur (e.g. 1(f) has involutions fixing 1 and 15 vertices) and are not classified here. Further results. For an automorphism of order 3, orbit counting allows f = 2, 5, 8, …, 38 fixed vertices; SAT (with the same symmetry breaking) excludes every case f = 5, 8, …, 38, and for f = 2 the two fixed vertices are necessarily adjacent. For a prime p ≥ 5, a non-identity automorphism with a fixed vertex would have to fix all four neighbours of that vertex (a non-fixed orbit of size p cannot fit among four neighbours), and by connectivity every vertex, a contradiction; so it is fixed-point-free, p divides 41, and p = 41, a case excluded by SAT. Subject to the SAT exclusion of the order-41 case, Cauchy's theorem therefore implies that |Aut| has the form 2^a·3^b for every (4,3) graph on 41 vertices (if a prime p ≥ 5 divided |Aut|, there would be an automorphism of order p).

Local completeness. Delete k ≤ 3 edges from one of the 16 graphs and pair the 2k exposed endpoints into k new edges (no loops or parallel edges): every result that is again 4-regular of diameter 3 is one of the 16 (SAT; reproduced by an exhaustive enumeration in separately written code).

Three closed families ("boxes"). Fixing a set of edges (a "core") and enumerating every way to complete it to a 4-regular graph of diameter 3 on 41 vertices (SAT, complete): a 70-edge core of N2 has exactly 64 labelled completions, which are Allwright's graphs 3, 4, 5 and 6, N1 and N2 (reproduced by separately written code: 64, with the same multiplicities 24/12/12/8/4/4); a 46-edge core from the single-swap closure of 2(c) has 147,456 completions by SAT, all isomorphic to 2(a), 2(b), 2(c) or N3 (multiplicities 73,728/24,576/24,576/24,576; separately written code counting embeddings of the core reproduces these four multiplicities, which by itself does not establish completeness); a 72-edge core common to 1(a)–1(f) has completions exactly 1(a)–1(f). So the 16 graphs lie in three closed families of 6, 4 and 6.

Additional symmetries. Besides the symmetries reported by Allwright for 1(a), 1(f), 2(a) and graphs 4, 5 and 6, we find involutions of 1(b), 1(c) and 1(e), and automorphism groups of order 6 for 2(b) and 2(c). These are among the symmetries the analysis above builds on.

Limits. The symmetry-enumeration statements above are computer results that rely on our SAT encodings and their symmetry breaking; they are cross-checked by software as described, but they have not been formally verified or checked by an outside expert. The existence of a 17th (4,3) graph on 41 vertices with none of the classified symmetries and outside the three families is not excluded.

The search programs, the SAT encodings, the cross-checks and this description were produced with substantial help from AI assistants (Anthropic's Claude; cross-checks with OpenAI's ChatGPT) under the author's direction. The verification described above is computational (separate programs and standard tools such as nauty); it has not been reviewed by an outside expert.