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Graph data mining algorithms rely on graph canonical forms to compare different graph structures. These canonical form definitions depend on node and edge labels. In this paper, we introduce a unique canonical visual matrix representation that only depends on a graph's topological information, so that two structurally identical graphs will have exactly the same visual adjacency matrix representation. In this canonical matrix, nodes are ordered based on a breadth-first search spanning tree. Special rules and filters are designed to guarantee the uniqueness of an arrangement. Such a unique matrix representation provides persistence and a stability which can be used and harnessed in visualization, especially for data exploration and studies.