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Walks, trails, paths, cycles, and circuits in a graph are sequences of vertices and edges with different properties Results about circuit in the context of graph theory can be found here. Some allow repetition of vertices and edges, while others do not.

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By analyzing the available paths of a graph, we can draw some conclusions Some sources refer to a circuit as a closed trail A circuit is a sequence of adjacent nodes starting and ending at the same node

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So if a trail starts and ends at the same vertex, and has length 3 or greater, then it is a circuit

Let's look at an example of a circuit. A circuit is a path that begins and ends at the same vertex Notice that a circuit is a kind of path and, therefore, is also a kind of walk We will use the graph below to classify sequences as walks, paths or circuits

Euler paths and circuits are the most fundamental concepts in graph theory Bcdb is the simple circuit Ok, thank you i think this helped cleared up the confusion A simple circuit is one of the sort $v_1, \dotsc, v_n, v_1$ where $v_i \neq v_j$ if $i\neq j.$ as pointed out in the comments, we also want $n>2$ above

In an undirected graph you'll also want to have a condition that excludes $v_1, v_2, v_1$.

A circuit is a closed trail That is, a circuit has no repeated edges but may have repeated vertices An example of a circuit can be seen below Notice how there are no edges repeated in the walk $hbcdefcgh$, hence the walk is certainly a trail

Additionally, the trail is closed, hence it is by definition a circuit. What is a circuit in graph theory It’s a question that unlocks a fascinating world of interconnectedness, where paths and cycles weave intricate patterns Understanding circuits is fundamental to graph theory, impacting diverse fields from network design to social network analysis.