Neighborhood of a Vertex | Open and Closed Neighborhoods, Graph Theory

What is the neighborhood of a vertex? Remember that the neighbors of a vertex are its adjacent vertices. So what do you think its neighborhood is? We’ll be going over neighborhoods, both open neighborhoods and closed neighborhoods, and an alternative definition of neighborhood, in today’s video graph theory lesson!
The neighborhood of a vertex v is the set containing all vertices adjacent to v. This is also called an open neighborhood. Unless otherwise stated, a neighborhood is assumed to be open. The closed neighborhood of a vertex v is the set containing all vertices adjacent to v as well as v itself. So if v is adjacent to a, b, and c, then the open neighborhood of v, written like this: N(v), is {a, b, c} and the closed neighborhood of v, written as N[v], is {a, b, c, v}.
It follows from the definition that the cardinality of the open neighborhood of a vertex is equal to the vertex’s degree.
Alternatively, the neighborhood of a vertex v may be defined as the subgraph induced by the set of all vertices adjacent to v. This definition extends similarly to closed neighborhoods. Watch the full lesson for more details!
SOLUTION TO PRACTICE PROBLEM:
The neighborhood of a is the set of all vertices adjacent to a, which is {c, d, f}. Remember we assume that the open neighborhood is desired since the question didn’t specify otherwise. The closed neighborhood of a, if desired, is {c, d, f, a}.
If you're taking a course in Graph Theory, or preparing to, you may be interested in the textbook that introduced me to Graph Theory: “A First Course in Graph Theory“ by Gary Chartrand and Ping Zhang. It’s a wonderful text! You can purchase this book through my Amazon affiliate link below! Using the affiliate link costs you nothing extra, and helps me continue to work on Wrath of Math!
PURCHASE "A First Course in Graph Theory": amzn.to/31hgvvJ
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.
Vallow Bandcamp: vallow.bandcamp.com/
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Vallow SoundCloud: / benwatts-3
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[@nastaransf]

N(a) = {c,d,f}, |N(a)| = deg(a)
N[a] = {c,d,f} U {a} , |N[a]| not= deg(a)
Thank you!

[@A-arons_mclovinit]

Really good for getting into the basics of neighborhoods

[@WrathofMath]

Glad it helped, thanks for watching! If you're looking for more graph theory, check out my playlist! th-cam.com/play/PLztBpqftvzxXBhbYxoaZJmnZF6AUQr1mH.html

[@verduregreen2991]

Thank you for this😊

[@WrathofMath]

You're very welcome! Thanks for your support!

[@atrsankaralingamvalarmathi3]

Excellent explanation Sir...Can u plz make some videos regarding algebraic graphs...it will be more helpful... Thank u Sir

[@garimakhetawat7220]

Nice explanation sir. Just an observation, please verify - If G is an empty graph and u is any of it's vertex, then by the 1st definition, N(u) is the empty set and N[u] is {u}. By the second definition, N[u] is the trivial subgraph consisting only the vertex u.
Question- How can we define N(u) by the second definition?
Another question- If G is the trivial graph, how can we define N(u)?

[@javadsa9300]

Thank you very much.

[@WrathofMath]

Glad to help, thanks for watching! Let me know if you have any questions, and check out my graph theory playlist if you're looking for more! th-cam.com/play/PLztBpqftvzxXBhbYxoaZJmnZF6AUQr1mH.html

[@wakaharris9050]

It was really understandable. Also, is the closed and open neighborhoods discussed in this video the same or the different to proper and improper neighborhoods? Thank you very much.

[@WrathofMath]

Thanks for watching! Unfortunately I am not familiar with the terms improper or proper neighborhoods. Where have you seen them?

[@dvendator]

Thankyou!

[@WrathofMath]

My pleasure! Thanks for watching!

[@moayyadarz2965]

That s really very good

[@WrathofMath]

Thanks for watching, I’m glad it helped!

[@user-pw5lh6ls1d]

شرح ممتاز عاشت ايدك

[@aboalesaad]

I like the software you're using. What is it ?

[@WrathofMath]

Notability for iPad!

[@abdullahakcay5770]

how about loops? if a had a loop, the card of n{a} = 2 but deg{a}=4

[@WrathofMath]

Thanks for watching and good question! In general, unless I say otherwise in a lesson, I am only talking about simple graphs. But if you're talking about the vertex a in the first graph of this lesson, if it were to have a loop then its neighborhood would have 3 vertices, { a, g, f } and its degree, like you said, would be 4.

[@rajeshwarishibaraya8462]

what is {neighborhood of vertex x }-x

[@WrathofMath]

Thanks for watching and you mean what do we call the neighborhood of x, but without x itself? In my experience, the neighborhood of a vertex doesn’t usually include the vertex itself - only the neighbors. You might call such a neighborhood an “open” neighborhood. Or, if you DID want it to include the vertex itself, you’d call it a “closed” neighborhood. Of course, if self loops are allowed, then these terms become a bit meaningless since the vertex could be its own neighbor.

[@afsahemaan9867]

N(a)={c,d,f}
N[a]={a,c,d,f}

[@PunmasterSTP]

Edges fall across the land,
The vertex hour is close at hand.
Math teachers scrawl, in search of blood,
To terrorize y'all's neighborhood.
😎