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

Yup!

iPad

Thanks for watching, Garima! Good question, I think I meant to make a lesson on that a while ago. Cycles and circuits are different because circuits are allowed to repeat vertices whereas cycles are not. Neither cycles nor circuits can repeat edges. If you look at a circuit, you can imagine it being composed of cycles that have single vertices in common, where the circuit repeats one. Does that help?

@@WrathofMath Yes sir, got it. Thankyou

@@WrathofMath at 5:30 v7 is repeated in the cycle, so does this make it a circuit.

Thanks for watching and for the question! I say in this lesson that we CANNOT repeat vertices in a cycle, with the exception of the first and last vertex, which are the same. Note that, most of the time, which vertex we call the first and last is arbitrary. So the first cycle I write out in this video (v2, v3, v4, v7, v2) could just as easily be written as (v3, v4, v7, v2, v3). Both are referring to travel across the same vertices and the same edges. Though, in that sequence representation, they are still distinct. But if you were to say - count the cycles in a graph - you probably wouldn't want to count both of those as separate, as they refer to the same substructure of the graph. Does that make sense? A circuit is a closed trail. That is, it is a walk that repeats no edges, and that begins and ends with the same vertex. Unlike cycles, circuits can repeat vertices internally as well, not just at the start and end. If you look at some circuits, you may notice that they can be broken into cycles.

Haha, if it doesn't it should only be a minor inconvenience. Some of these graph theory videos are starting to approach 5 years old which is nuts, but I believe most of my terminology should align with A First Course in Graph Theory by Chartrand and Zhang, since that's the text most of the playlist is based on. Having used the terminology and definitions from that book, I have not found the wider graph theory literature to be in much conflict with it.

Thank you! I do love the colors and the look of the writing in the software I use. Big step up from the old videos in appearance.

no

You're very welcome, glad it helped and thank you for watching!

Thanks for watching and for the question! Always be wary of the definition being used, I believe some authors use "cycle" to refer to a closed walk that repeats no edges (but can repeat vertices) and use "simple cycle" to refer to a closed walk with no repeat vertices (aside from the first and last). But what I find is most common, and what I use in my graph theory videos, are the following definitions: cycle: closed walk repeating no vertices (aside from first and last) circuit: closed walk repeating no edges Does that help? Here is my lesson on circuits: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-_YvlvSO7RBQ.html

@@WrathofMath wow i never expected you will reply sir , i am actually binge watching your play list can you give more information on books to learn about Graphs in terms as Algorithms thanks alot

You're very welcome! Thanks for watching! Let me know if you ever have any video requests!

Haha - I guess I used Vallow music at the end of this video, he goes by "Crayon Angel" now if you ever want to listen to his music. The song I used in the sped up part, and at the end of some other videos, is my song, which I never released, but some of my music is available for channel members. Thanks for watching!

@@WrathofMath That's awesome, and it's great to hear from you! Thanks again for putting together such an amazing playlist.

Thank you!

My pleasure, thanks for watching and let me know if you ever have any questions!

Thanks for watching! That's a good question. It depends exactly what you want to learn about and what you already know. If you aren't too familiar with Graph Theory, I'd begin with an undergraduate Graph Theory textbook. If you check the description of any of my recent graph theory lessons, you'll see an affiliate link to the textbook that introduced me to Graph Theory, called "A First Course in Graph Theory" by Chartrand and Zhang. It's a very affordable $15 or so on Amazon. It also includes several appendices that will help freshen you up on relevant material you need for the text, like set theory, logic, functions, and proofs. There are also plenty of free online resources for Graph Theory. I have plenty more lessons on my channel, and plenty more coming, but they are not in the structure of a course (though this is something I will provide in the future, and look forward to that mightily!). Sarada Herke created a RU-vid channel with tons of wonderful graph theory videos, following the structure of a course. If you're more interested in cycles specifically, I would look up some papers on problems concerning cycles that you have enough background to understand with some effort, which will deepen your understanding and curiosity. Good luck in your studies!

thank you

Appreciate you! Have fun!

It is also a cycle....it is satisfying the nasic definition of cycle,right!

Thank you! Let me know if you ever have any video requests!

That depends what you mean! A "cyclic" graph is defined to be a graph containing a cycle, so there isn't much need to talk about that definition in a dedicated lesson. But if you mean to ask if I have a lesson on cycle graphs, it is here: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-tcCoQySN7Xs.html

Thank you sir 👍🏽

Thanks for watching, Rohan! That song is one I wrote called "Turn Over". Unfortunately it was never finished and posted anywhere except for the handful of math lessons I used it in (usually at the end). However I do have other music posted on this channel: ru-vid.com/show-UCBDXtKCGkvF-bWfuf6JNDiQ And this is my new music channel I intend to use going forward: ru-vid.com/show-UCOvWZ_dg_ztMt3C7Qx3NKOQ

@@WrathofMath you must complete that song. It sounds nice in the the end of the video

Thanks for watching and absolutely! I assume you mean the cycle described by this vertex sequence: (v7, v4, v5, v6, v7). That's a 4-cycle, sometimes called a square.

@@WrathofMath Thank you. BTW, how to determine whether two graphs are isomorphic by counting “cycle”. :)

this question is in my mind also any body clear me