[Math] what is the difference between a cycle and a circuit in graph theory

graph theory

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.