It can be very hard to see that two graphs are equivalent just by inspecting the diagram. Also, at times two graphs can appear to be equivalent but actually be different. There is a method to determine if two graphs are equivalent using their adjacency matrices. Even two equivalent graphs can have different adjacency matrices. However, interchanging vertices on the graph is equivalent to interchanging rows and columns of the matrices.
Lets consider an example with two graphs.
This means that these two graphs are actually equivalent. If you look very long and hard at the matrices you can sometimes see which rows and columns to swap. We can create a little Maple procedure that will check all of the possibilities and then tell us if the matrices are equivalent, and if so, which vertices need to be swapped. In all of the graphs we ve seen so far, it has been possible to get from any vertex to another by travelling one or more edges. This type of graph is called a connected graph.