We introduce eigencone constellations, a hierarchical framework for embedding bounded-degree spatial graphs into concentric spherical shells and partitioning each shell into spectrally weighted, spherical star-shaped territories. Given a connected sparse spatial graph $G$ with a distinguished root vertex (the queen), we assign each vertex to a sphere whose radial position is determined by its graph distance from the queen, then tessellate each sphere into constellation territories whose solid angles are proportional to the spectral mass of the corresponding subgraph. Within each territory, nodes are packed by constrained repulsion, yielding local simplex structures. The resulting geometric representation provides a structural framework for measuring spectral distance between dynamic subgraph states. By combining this eigencone-derived metric with constraints on the domain-specific edit alphabet, we define a forward-only deterministic trajectory -- the isomorphic walk -- which converges graph edits efficiently. We define the notion of spherical star-shaped domains with geodesic visibility, establish their properties under spectral projection, and demonstrate the trajectory convergence on molecular contact graphs.

翻译：我们提出特征锥星座结构，这是一种层级化框架，用于将有界度空间图嵌入同心球壳，并将每个球壳划分为谱加权的球面星形领地。对于给定连通稀疏空间图$G$及其指定的根顶点（蜂后），我们将每个顶点分配到由该顶点与蜂后之间图距离决定径向位置的球面上，然后将每个球面分割成星座领地，其立体角与对应子图的谱质量成正比。在每个领地内，通过约束排斥机制实现节点的紧密排列，从而形成局部单纯复形结构。这种几何表示法为测量动态子图状态间的谱距离提供了结构框架。通过将该特征锥派生度量与领域特定的编辑字母表约束相结合，我们定义了一个前向确定性轨迹——同构游走——能够高效收敛图的编辑操作。我们定义了具有测地线可视性的球面星形域概念，建立了其在谱投影下的性质，并在分子接触图上验证了轨迹收敛性。