In symbols: This is referred to as Lo's condition in the literature. This follows from the fact that the sum of the labels of the vertices is twice the sum of the edges, modulo . This is useful for disproving a graph is edge-graceful. For instance, one can apply this directly to the path and cycle examples given above. Further selected results The Petersen graph is not edge-graceful. The star graph (a central node and m legs of length 1) is edge-graceful when m is even and not when m is odd. The friendship graph is edge-graceful when m is odd and not when it is even. Regular trees, (depth n with each non-leaf node emitting m new vertices) are edge-graceful when m is even for any value n but not edge-graceful whenever m is odd. The complete graph on n vertices, , is edge-graceful unless n is singly even, . The ladder graph is never edge-graceful. References Graph theory objects