发表机构
School of Mathematics and Statistics, Nantong University（南通大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文提出一种精确递推算法，通过逐循环处理置换并仅记录剩余循环长度和度数，实现无标号四次图的高效计数，达到指数级复杂度上界，并验证了至阶数50的计数结果。
AI 中文摘要
无标号正则图的数量可以表示为顶点置换上不动点计数的平均值，但评估每个不动点计数仍需强制满足度约束。我们给出一个精确递推，每次处理一个完整的置换循环，并仅通过其长度和剩余度数记录其余循环。该递推将内部边轨道与连接不同循环的轨道相结合，而二项式系数和多项式系数保留了导致相同剩余状态的选择多重性。我们证明在完全循环消除下该状态描述是充分的，并推导出状态数和转移数的界限。对于每个固定度数，所得算法在顶点数上具有 $\exp(O(\sqrt n))$ 的上界，包括整数算术成本。四次情况每个循环长度仅需四个正剩余度数类别。与单独实现的、基于顶点索引的边轨道计算的小实例比较验证了正则和非均匀剩余度数输入。四次计算给出直到阶数50的无限制和连通计数，包括超出相应参考表（至阶数28）的22个阶数。连通计数通过逆欧拉变换恢复，且阶数29--50的所有22个恒等置换贡献与已发表的标号计数一致。这些阶数上的非恒等不动点项尚未被独立重新计算。
英文摘要
The number of unlabeled regular graphs can be expressed as an average of fixed-point counts over vertex permutations, but evaluating each fixed-point count still requires the degree constraints to be enforced. We give an exact recurrence that processes one complete permutation cycle at a time and records the remaining cycles only by their lengths and residual degrees. The recurrence combines internal edge orbits with orbits joining distinct cycles, while binomial and multinomial coefficients retain the multiplicities of choices that lead to the same remaining state. We prove that this state description is sufficient under complete-cycle elimination and derive bounds on the number of states and transitions. For every fixed degree, the resulting algorithm has an $\exp(O(\sqrt n))$ upper bound in the number of vertices, including integer-arithmetic costs. The quartic case requires only four positive residual-degree classes for each cycle length. Small-instance comparisons with a separately implemented, vertex-indexed edge-orbit calculation verify both regular and nonuniform residual-degree inputs. The quartic calculation gives unrestricted and connected counts through order 50, including 22 orders beyond the corresponding reference tables through order 28. Connected counts are recovered by the inverse Euler transform, and all 22 identity-permutation contributions for orders 29--50 agree with the published labeled counts. The nonidentity fixed-point terms at these orders have not been independently recomputed.
