Cross-file mathjax reference: \ref{blatzo}

We might rightly begin an exploration of graph theory with topics grouped together with a common theme of connectivity.

There is often considerable confusion about these terms. Consider the following definition.

A

Suppose that

Induction on

So this definition does not preclude visiting

a vertex more than once, nor going back-and-forth along a single edge.

This is one of my favorite topics.

Might as well begin here.

We can have a variety of displayed equations, basically from the amsmath package. For example, a single displayed equation, with no number, and hence no referencing is possible:

We can can have a single numbered equation, so referencing is possible:

Or several equations in a row, none of them numbered.

Or several equations in a row, all of them numbered.

We can selectively not number equations in a group that are all numbered.

We can selectively number equations in a group that are all unnumbered.

We should first make a definition.

Given a graph

When no confusion will result, we will denote this matrix simply as

This is where the fun would start. But instead we will practice referencing some displayed equations, such as the first numbered single display, Equation~

Test refs: \ref{blatzo}, \ref{incompatible}

One of the best things about graph theory is that you can draw pictures. Here is a classic.

I also like Heawood's graph.

Regular graphs have a certain amount of combinatorial symmetry.

We call a graph

Very pretty, no?

Let's reference a previous definition on walks: Definition

And a reference to the first Heawood graph: Figure

There is a nice proof by induction at