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.
Definition 1.1 A walk in a graph is a sequence of vertices such that consecutive vertices in the sequence are adjacent in the graph.
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 should first make a definition.
Definition 1.1 Given a graph \(G\), we define the adjacency matrix \(A(G)\) as the \(0\)-\(1\) matrix given by: \[A(G)=\begin{cases} 0 &\text{ if }v_i\text{ and }v_j\text{ are adjacent}\\ 1 &\text{otherwise} \end{cases}\]
When no confusion will result, we will denote this matrix simply as \(A\).
Sample notation (in a master list eventually): \(A(G)\)
This is where the fun would start.