# Vertices

**Introduction:** A ** graph** is a non-empty finite set

`V`of elements called

**together with a possibly empty set**

*vertices*`E`of pairs of vertices called

**edges**.
**Degree of a Vertices**: The number of edges incident on a vertex vi with self-loops counted twice (is called the degree of a vertex vi and is denoted by degG(vi) or deg vi or d(vi). The degrees of vertices in the graph G and H are shown in Figure 4(a) and 4(b).

In G as shown in Figure 4(a),

**degG (v2) = 2 = degG (v4) = degG (v1), degG (v3) = 3 and degG (v5) = 1 and**

In H as shown in Figure 4(b),

**degH (v2) = 5, degH (v4) = 3, degH (v3) = 5, degH (v1) = 4 and degH (v5) = 1.**

The degree of a vertex is some times also referred to as its valency.

**Isolated and Pendent Vertices:
**

**Isolated vertex**

A vertex having no incident edge is called an isolated vertex. In other words, isolated vertices are those with zero degree.

**Pendent or end vertex**

A vertex of degree one, is called a pendent vertex or an end vertex. In the above Figure, v5 is a pendent vertex.

**In degree and out degree**

In a graph G, the out degree of a vertex vi of G, denoted by out degG (vi) or degG^{ } (vi), is the number of edges beginning at vi and the in degree of vi, denoted by in degG (vi) or 1 degG^{−1} (vi), is the number of edges ending at vi.

The sum of the in degree and out degree of a vertex is called the total degree of the vertex. A vertex with zero in degree is **called a source** and a vertex with zero out degree is **called a sink.** Since each edge has an initial vertex and terminal vertex.