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- Understanding Eigenvalues of Path and Complete Bipartite Graphs | Spectral Theory | Fall 23
Understanding Eigenvalues of Path and Complete Bipartite Graphs | Spectral Theory | Fall 23
Explore the symmetry of eigenvalues in path and complete bipartite graphs in spectral theory. Learn about the significance of eigenvalues and their applications in group-generated graphs such as Kelly graphs. Dive into the concept of generators adding or removing elements in group structures for circular graphs.
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Video Transcript
Last time, we talked about the eigenvalues of a few graphs.
We were able to figure out,
we started with the empty graph,
and then of course naturally jumped to the cycle,
which took us on a half hour detour.
But of course, the cycle is just a really special case
of circulant graphs, and these show up in a lot of places.
And in particular, when you look at graphs
which arise from groups.
So there's something called Cayley graphs, where you have the elements of your group
are your vertices and then certain actions act on it.
So if your group happens to be the group called Zn, the numbers modulo n, then you can talk
about generators are adding or subtracting by some number, some subset.
And then what you get are circular graphs.
So okay, all right, so we've done these.
So I have...
some things I'd like to do and we'll see how far I get because you know I
ramble so but that's okay. We have a whole semester there's there's no rush
no rush at all. Alright so are there any other graphs that you want to try?
Well, we didn't solve paths, but we can do paths.
So okay, so we can talk about paths.
All right. Now, first off, we should all understand what the path looks like.
And so I'm going to say P sub n is the following graph.
I'm going to draw it visually where you have...
This is the part where there's controversy and vertices.
So that's p sub n.
And so you can label the vertices 1 through n.
And i is adjacent to i plus 1 for 1 less than or equal to i
less than or equal to n minus 1.
I think the picture is pretty convincing.
Now, the path, it's not quite so obvious how to start.
So I'm going to take a small detour.
And this will build off of things we talked about yesterday.
And all right, so let's recall that when we talk about eigenvalues, we should of course
mention we mean adjacency matrix.
And I know that that's the one we've defined.
You can imagine every time we introduce a matrix, there will be some activities like
this where we're like, okay, now what happened?
But, all right, so we're in the adjacency matrix.
We talked about...
the idea of the adjacency operator, which is to say that when we think of a vector,
it's just assigning values to the vertices. And when we apply the matrix multiplication by A,
that's the adjacency operator, you add the neighbors. So observation.
If for, and of course this takes a long time to say things,
and it's much fun when you can just wave your hands.
All right, so if for our graph G, adjacency matrix A,
so we're introducing our pieces here,
and let's say our eigenvector X for eigenvalue lambda.
Okay, so we have all of this stuff here.
At this point you're probably saying,
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Video Transcript
上次,我们谈论了一些图表的自我价值。
我们能够发现,
我们从空的图表开始,
然后自然地跳进了循环,
他们把我们带走了半个小时。
但当然,周期只是一个非常特殊的案例。
流动图表,这些在许多地方出现。
特别是当你看图表时
是由群体产生的。
所以有一件事叫做凯利图,在那里你有你的团体的元素。
这是你的顶部,然后某些行动在它上发挥作用。
所以,如果你的群体是称为 Zn的群体,数字模块 n,那么你可以说话。
关于发电机正在添加或取消某些数字,某些子组。
然后你得到的是圆形图表。
好吧,好吧,所以我们做了这些。
所以我有一些我想做的事情,我们会看到我有多远,因为,你知道,我正在奔跑。
所以,但这很好,我们有一个整个学期,没有匆忙,没有匆忙。
好吧,那么还有其他图形你想尝试吗?
好吧,我们没有解决路径,但我们可以做路径。
所以,好吧,我们可以谈论道路。
全都好。
首先,我们都应该明白这条路是什么样子。
因此,我会说 P sub n 是下一个图表。
我将视觉地绘制它在你有的地方,这是争议的部分,
n verticals. 因此,它是 p sub n. 因此,你可以标记的 verticals 1 通过 n, 和 i
与 i 加上 1 对 1 较少或等于 i 较少或等于 n minus 1 。
我认为这张照片相当令人信服。
现在,路线,不太清楚如何开始。
因此,我要做一个小分散。
这将建立在我们昨天谈论的事情上。
是的,所以让我们记住,当我们谈论自己的价值观时,我们当然应该
我们说的是Ajacency Matrix。
我知道这就是我们所定义的唯一一个。
你可以想像每當我們引入一個 matrix,
会有一些这样的活动。
我们像,好吧,现在发生了什么?
但是,没错。
因此,我们正处于邻近地图。
我们谈论了邻居运营商的想法,
也就是说,当我们想起一个引擎时,
它只是将价值分配给顶部。
當我們用 A 加倍數時,
这就是邻居运营商,你添加邻居。
所以观察,如果是,当然,这需要很长时间。
说话,这是很有趣的。
只要你能伸出手。
全都好。
因此,如果对于我们的图形G,接近 matrixA,
所以,我们在这里介绍我们的作品,让我们说我们的自导体 x为自导品 lambda。
好吧,所以我们在这里所有这些东西。
