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PROFESSOR: In this
video lecture,

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we're going to introduce
the idea of directed graphs,

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or digraphs for short.

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So normally and
before this class,

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you might have thought of graphs
as being something like this.

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Y is a function of x and
graphed on the xy plane.

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But that's not what we
want to be thinking about.

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Instead, we want to think
about something like this.

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This is a graph to a
computer scientist.

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Show a bunch of vertices, which
are those point that you see,

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and edges, which
connect vertices.

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Being more specific
and direct about this,

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it's composed of a
set V of vertices

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and a set E of edges, which
are composed of 2V each.

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The way you write that
out, an edge is v comma w,

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and that specifies an
edge going from v to w.

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And in the graph, it would
look something like this.

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Note that they are directed.

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That an edge from v to
w is not the same thing

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as an edge from w to
v in a directed graph.

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For example, here we
have one directed graph,

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and you write out vertices as
the set of all the vertices

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you see there.

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And edges are pairs of vertices.

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You can also
realize that digraph

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is the same thing as a binary
relation on the vertices,

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because each edge just
defines the relationship

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from one vertice to another.

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So, every binary relation
can be drawn out as a graph.

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You just put each of the things
in each of the sets as vertices

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and edges being the
things that relate

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from one set to the other.

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So, now we're going to
define walks and paths.

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Now, a walk follows
successive edges

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but it can repeat
vertices or edges.

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For example, I can start
at the black vertice there,

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and we can go to red,
blue, yellow, red.

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And w can go back to blue again.

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There's nothing stopping us.

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And the length of
these paths is not

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how many vertices
we've gone through,

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but the number of edges
that we've gone through.

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So here, the length
would be five

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because we went from white
to black, black to blue,

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blue to yellow,
yellow red, red blue.

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It's not the six vertices
that we went through.

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And you have to be
careful about that,

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because that difference
of one kind of gets you.

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A path, on the other hand,
walk through vertices with that

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repeating a single vertex.

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So, for example, start at
blue, you can go to yellow,

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you can go red, you can
go pink, you can go green,

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but then we're stuck.

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We can't go back to red.

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We've already been there.

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So, that's it.

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That would be the
end of our path.

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If we went to red again, then
it wouldn't be a path anymore.

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Not be a valid path.

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And here, the [INAUDIBLE]
length is four edges, not

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five vertices.

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And every graph
can be represented

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as a matrix representation.

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You draw it out like this.

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And what we're going
to do is if there's

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a edge that goes from one of the
things on the right over to one

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of things on the top, we'll
put a one at that position.

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For example, there's an edge
that goes from the black

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to the red.

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So, on the black row
in the red column,

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we're going to put in a one.

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Same thing, there's one that
goes from black to green.

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We'll put black row, green
column, put in another one.

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And so on for all the edges
that we have in our graph.

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And the rest we just
filled with zeroes.

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And this is called
an adjacency matrix.

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And as you can see, it
uniquely defines a graph.

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Every edge is represented here,
and every one of the vertices

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is represented here.

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So, any graph can be
drawn up this way.