Vectorial Graph Distance

We introduce a Vectorial Graph Distance (VGD), a distance measure between graphs, a novel metric that extends our previously defined Vectorial Tree Distance (VTD) to general undirected, unweighted graphs. The method converts each graph into a unique tree by combining a bounded quotient graph, constructed using node-betweenness centrality, with a modified breadth-first search (BFS) that ensures uniqueness. Once converted, the resulting trees are compared using the VTD, yielding a structured, vector-valued distance that reflects hierarchical differences between graphs. We further demonstrate the utility of the VGD by applying it to the ENZYMES dataset (BRENDA), showing that it successfully distinguishes molecular graphs from different enzyme classes. The approach captures meaningful structural distinctions and provides an interpretable, multi-level measure of graph similarity.