Isomorphism Determination Methods for Simple Undirected Graphs

This work gives the definitions of matrix isomorphic transformation, distance matrix of the simple undirected graph, column sum vector of the distance matrix, and distance spectrum of the graph, which extend the adjacency matrix-based isomorphism determination conditions to the distance matrix-based ones for simple undirected graphs. For the isomorphism determination problem of simple undirected connected graphs: One determination condition based on the characteristic polynomial of distance matrix is proposed; Further, another determination condition based on the rank and the column sum vector of distance matrix is proposed to avoid the influence of calculation error on the determination result. These two determination conditions are both necessary and sufficient conditions and both have polynomial time complexity. © 2023 Science Press. All rights reserved.