Isolated Vertices, Leaves, and Pendant Edges

Isolated Vertices

Definition: For a graph $G = (V(G), E(G))$, a vertex $x_1 \in V(G)$ is considered Isolated if $\mathrm{deg} (x_1) = 0$.

For example, the following graph has one isolated vertex:

Note that if a graph has an isolated vertex, then the graph is disconnected.

Leaves

Definition: For a graph $G = (V(G), E(G))$, a vertex $x_1 \in V(G)$ is considered a Leaf if $\mathrm{deg} (x_1) = 1$.

For example, the following graph has one leaf, namely the vertex labelled "$1$":

Pendant Edge

Definition: For a graph $G = (V(G), E(G))$, an edge connecting a leaf is called a Pendant Edge.

From the example earlier, we can highlight the pendant edge of the graph: