Counting non-isomorphic tree-like multigraphs that include self-loops and multiple edges is an important problem in combinatorial enumeration, with applications in chemical graph theory, polymer science, and network modeling. Traditional counting techniques, such as Polya's theorem and branching algorithms, often face limitations due to symmetry handling and computational complexity. This study presents a unified dynamic programming framework for enumerating tree-like graphs characterized by a fixed number of vertices, self-loops, and multiple edges. The proposed method utilizes canonical rooted representations and recursive decomposition of subgraphs to eliminate redundant configurations, ensuring exact counting without the need for explicit structure generation. The framework also provides analytical bounds and recurrence relations that describe the growth behaviour of such multigraphs. This work extends previous models that treated self-loops and multiple edges separately, offering a general theoretical foundation for the enumeration of complex tree-like multigraphs in both mathematical and chemical domains.

翻译：对包含自环与多重边的非同构树状多重图进行计数是组合枚举领域的重要问题，在化学图论、高分子科学和网络建模中具有应用价值。传统计数方法（如Pólya定理和分支算法）常因对称性处理与计算复杂度的限制而面临挑战。本研究提出一种统一的动态规划框架，用于枚举以固定顶点数、自环数和多重边数为特征的树状图。该方法利用规范有根表示与子图的递归分解来消除冗余构型，确保无需显式生成结构即可实现精确计数。该框架还提供了描述此类多重图增长行为的解析界与递推关系。本工作扩展了以往将自环与多重边分别处理的模型，为数学与化学领域中复杂树状多重图的枚举提供了普适的理论基础。