Consider the graph with vertex set {A, B, C, D, E} and edge list A C, A E, B D, B E, C A, C D, C

Consider the graph with vertex set {A, B, C, D, E} and edge...

Key Concepts

Graph

A graph is a collection of objects called vertices that are connected by edges. In graph theory, these vertices and edges are used to model pairwise relationships between objects. The focus is on the structure of the graph rather than the specific labels of vertices or the positions in which they are drawn.

Graph Representation

Graphs can be represented in various ways, including edge lists, matrices, and visual drawings. An edge list expresses a graph by listing its individual edges as pairs of vertices, while drawings visually illustrate the relationships between vertices. Both representations capture the same underlying structure of the graph.

Graph Drawing and Visualization

Drawing a graph involves positioning the vertices and drawing lines for the edges in the plane. Different drawings of the same graph can highlight different properties of the graph, such as symmetry and connectivity, and can be useful in understanding the structure and properties of the graph, even if the drawings look visually distinct.

Graph Isomorphism

Graph isomorphism is the concept that two graphs can be structurally identical even if they are drawn differently. Different pictures of the same graph, though they may look dissimilar, represent the same abstract graph if there exists a one-to-one correspondence between their vertices that preserves adjacency.

Transcript

00:01 So for problem number 56, we're looking at two situations addition of complex numbers.

00:10 And they're keeping this in their abstract forms.

00:14 So no concrete values.

00:18 And we're looking at the sum of complex numbers a plus bi and a minus bi compared with the graph of the difference.

00:34 Of these two complex numbers.

00:39 And hopefully we recognize that when we've got these opposite signs here, that these are called complex conjugates.

00:57 Complex conjugates, when you change the sign of the imaginary part.

01:03 So let's think back to when we are adding and subtracting.

01:11 Complex numbers.

01:14 It's almost like vectors here.

01:16 So if i plot the point a plus bi, right? that means i have maybe a positive real number.

01:25 And we're graphing this in the complex plane and a positive imaginary.

01:32 So maybe here is where that point a -b is for that a -plus b.

01:40 What's going to change is if we have the imaginary part negative is that's going to be the same distance a in the real direction but now we're going to be reflected across that real axis and we're going to be here okay so let's think about this in terms of maybe some vectors the modulus of each value there each complex number from the origin out to that point.

02:15 Now, if i have two vectors like this and i want to add them together, remember i would have to place these head to tail...

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