AN INVARIANT OF THE GRAPH ISOMORPHISM

alpha The adjacent matrix is defined in this paper for a given graph, and the deformation called dual congruent act on the matrix is discussed. We have shown that any two graphs are isomorphic each other if and only if they having the dual congruent adjacent matrices. So the adjacent matrix of graph is a isomorphic invariants for graph theory. Some properties about adjacent matrix are obtained in the paper that is for any graph G with vertices {v(1), v(2), ... , v(n)), the ith row sum of the adjacent matrix defined on G is equal to deg(v(i)). If two adjacent matrices are dual congruent, then they have the same eigenvalues and the determinants of matrices. But having the same determinants is only a necessary condition for graph isomorphism. The congruent matrix is a mathematical conception, it is easy to see that if the numbers of vertices of graph is n, then the simplest adjacent matrix is as follow: diag(deg(v(1)), deg(v(2)), ... , deg(V(n))). With the matrix above, we can write any symmetric matrices and corresponding graph, so the no isomorphism graphs which have n vertices are infinity.