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Handout 5.1 Notation and Terminology
Peer instruction questions 1–3.
Activity 5.1.1.
Suppose
\(\bfH\) is an induced spanning subgraph of a graph
\(\bfG\text{.}\) Discuss with your group what this would mean.
Peer instruction questions 4–5.
On the Complexity of Graph Isomorphism
Theorem 5.1. The First Theorem of Graph Theory.
Let \(\deg_{\bfG}(v)\) denote the degree of vertex \(v\) in graph \(\GVE\text{.}\) Then
\begin{equation*}
\sum_{v\in V}\deg_{\bfG}(v) = 2|E|\text{.}
\end{equation*}
\begin{equation*}
|E| = \frac{1}{2} \sum_{v\in V}\deg_{\bfG}(v)
\end{equation*}
Corollary 5.2.
In any graph, the number of vertices of odd degree is even.
Definition 5.3.
Theorem 5.4.
Every tree on at least two vertices has at least two leaves.
Theorem 5.5.
If
\(\bfT\) is a tree, then for every pair of distinct vertices
\(u,v\text{,}\) there exists a unique path from
\(u\) to
\(v\) in
\(\bfT\text{.}\)
Activity 5.1.2.
-
With your neighbors, use mathematical induction to prove that every tree on \(n\) vertices has exactly \(n-1\) edges.
-
How many edges would an \(n\)-vertex forest consisting of \(k\) trees have?
