A cycle of a graph G, also called a circuit if the first vertex is not specified, is a subset of the edge set of G that forms a path such that the first node of the path corresponds to the last.

This is the definition of cycle in the WolframMathWorld page. I don't understand how a cycle becomes a circuit if the 1st vertex is not specified.

I thought :

a circuit is a closed walk with no repeated EDGES.

a cycle is a closed walk with no repeated edges and no repeated vertices

## Best Answer

In graph theory conventions unfortunately differ in different contexts and with different authors. Your definition of cycle is the usual one except that cycles are often given with a specific order of vertex, edge, vertex, ... . Circuits can then be considered to be cycles but with no specific starting point.