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A matrix is a table that provides values for all row/column combinations. An adjacency matrix is a matrix that indicates whether features are or are not adjacent, or sharing edges.
While Esri does not have a tool that directly creates adjacency matrices, there is a tool called Polygon Neighbors. This tool creates a table with a row for every shared edge in a dataset. The tool can also output values such as the total length of the shared edge and the number of vertices shared. This helps a user understand if adjacent polygons are sharing just an edge, or just vertices (for example, a corner to a corner). The user can then use this output and create custom Python code to create an adjacency matrix as needed.