(15 points) Which of the following graphs are isomorphic. If...

Name: (15 points) Which of the following graphs are isomorphic. If graphs are isomorphic, establish an explicit isomorphism and prove that your function is indeed an isomorphism. If graphs are not isomorphic, give a proof as why not.
Uploaded: 2022-12-18T23:14:30-08:00
Duration: 4 min 39 s
Channel: Sydney Bement
Description: (15 points) Which of the following graphs are isomorphic. If graphs are isomorphic, establish an explicit isomorphism and prove that your function is indeed an isomorphism. If graphs are not isomorphic, give a proof as why not.

(15 points) Which of the following graphs are isomorphic. If graphs are isomorphic, establish an explicit isomorphism and prove that your function is indeed an isomorphism. If graphs are not isomorphic, give a proof as why not.

Transcript

00:01 Okay, so for this problem, we're given these two graphs and we're asked, are these isomorphic to each other? and if so, give the mapping.

00:12 So first things first, when we're dealing with isomorphisms, we need the number of elements in this one to equal the number of elements in this one.

00:22 And in this case, that means the nodes or the little dots.

00:27 So we have one, two, three, four, five, six little dots in our first picture.

00:36 Let's just go ahead and label this one and this two, just so we know what we're talking about.

00:41 Then we have one, two, three, four, five, six, and two.

00:48 So let's go ahead and do the cardinalities, both equal six.

00:53 So this does have the potential to be both bijective and one to one, or both surjective and one to one.

01:04 Or in other words, bijective, which is a great thing when we're being isomorphic to each other.

01:10 Now, in this case, we need the degree of each of these nodes, both the blue ones and the green ones, to have another node that has, the same degree.

01:26 So that's a little confusing.

01:29 So what i'm saying is if i have a node here and that has one line to, let me do it in a different color, one line to it, two lines and three lines.

01:41 So this has degree three.

01:45 So if i have a node in my map one that has degree three, i have to have a node of degree three in my map of or in my graph of two.

01:57 Let's look at a.

01:58 This has one, two, and three.

02:01 So this has degree three.

02:03 So we're all good there...