<p>Graph generation has attracted considerable attention in recent years. Most existing work focuses on generating graphs from collections, such as proteins, while the problem of generating a proxy from a single graph with comparable properties remains underexplored. Prior approaches often emphasize modifying neural architectures or training on edges with fixed node sets, which tends to produce high edge overlap and limited diversity. In addition, many neural graph generation methods overlook the mathematical structure of the input graph. For instance, key metrics such as wedge counts, which are determined by degree distributions, are often neglected. To address these limitations, we introduce a graph generation method that incorporates graph-theoretic principles into the learning process. Our approach preserves both global and local characteristics of the input graph while correcting the degree distribution to avoid duplicating the original topology. Unlike other single-graph methods that fix the node set, we generate new node embeddings and sample fresh node sets, yielding new connection topologies with substantially lower edge overlap.</p>