What is a Graph Cycle? | Graph Theory, Cycles, Cyclic Graphs, Simple Cycles

What is a graph cycle? In graph theory, a cycle is a way of moving through a graph. We can think of a cycle as being a sequence of vertices in a graph, such that consecutive vertices are adjacent, and all vertices are distinct except for the first and last vertex, which are required to be the same.

A cycle can be represented as a sequence of vertices in a graph, but it is sometimes also represented in other ways. For example, if ( a, b, c, d, a ) is a cycle in a graph, it could also be represented as a sequence of edges. We could write it as ( ab, bc, cd, da ). Additionally, a cycle is sometimes written as an alternating sequence of vertices and edges like this: ( a, ab, b, bc, c, cd, d, da, a ). These are just a few ways to write cycles.

The length of a cycle is the number of edges traversed in the cycle. So if a cycle is written as a sequence of vertices, the length of the cycle is one less than the number of vertices in the sequence. If the cycle is written as a sequence of edges then the length is just the number of edges in the sequence. The length of a cycle must be 3 or greater.

Sometimes cycles are defined slightly differently, so that vertices are allowed to be repeated but not edges. In this case, the cycles we previously discussed, that don't repeat vertices except for the first and last, are often called simple cycles.

I hope you find this video helpful, and be sure to ask any questions down in the comments!

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The outro music is by a favorite musician of mine named Vallow, who, upon my request, kindly gave me permission to use his music in my outros. I usually put my own music in the outros, but I love Vallow's music, and wanted to share it with those of you watching. Please check out all of his wonderful work.

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Support the production of this course by joining Wrath of Math as a Channel Member for exclusive and early videos, original music, and upcoming lecture notes for the graph theory series! Plus your comments will be highlighted for me so it is more likely I'll answer your questions!

[labiribiri]

So, a simple cycle is basically a closed path with an edge length of 3 or more?

[Notsorandomnumbers]

how many cycles does your example graph have? assuming you cannot repeat vertices

[luciano]

man, i hope the terminology in my class next semester aligns with yours haha

[jiangzzi]

A great lesson! May I know what tool (software) were you using for writing & drawing?

[SUPERDUPERPETS]

if your only given the set of the edges traveled how would you know if this is a cycle or circuit coz they both start and end with the same vertex ?

[manizhahashimi]

thank you for this beautifull playlist, can u please tell me the name of programm ur using for wrtting?

[wexwexexort]

are we sure that vertices can repeatin cycles? what's the difference between a cycle and circuit then?

[garimakhetawat]

Sir, do cycles and circuits represent the same thing?

[viktoriacsuka]

Thank you so much, you helped me a lot!

[pochengou]

can cycles repeat edge or nodes(vertex) ?

[ELOYGARCIA-cf]

then what´s the difference between a cycle and a circuit?

[bulidiriba]

thank you its an amazing explanation!!

[PunmasterSTP]

Whoa, it feels so strange to hear that music anywhere else besides the end of the video!

[AbirHasan-npru]

Sir, Why didn't it happen?like-(v2, v3, v4, v5, v6, v7, v2)

[Vr_rohan]

you haven't discussed what a cyclic graph is?

[talentmarandure]

Here in 2024 for the best Playlist on graph theory

[Vr_rohan]

Sir, please provide the name of the song that begins at 0:18.