Edges and Vertices of Graph

A graph is a set of points, called nodes or vertices, which are interconnected by a set of lines called edges. The study of graphs, or graph theory, is an important part of a number of disciplines in the fields of mathematics, engineering, and computer science.

Graph Definition

A graph (denoted as G = (V, E)) consists of a non-empty set of vertices (or nodes) V and a set of edges E. A vertex represents an endpoint of an edge. An edge joins two vertices and is represented by the set of vertices it connects.

Example

Let us consider a graph G = (V, E) where V = {a, b, c, d} and E = {{a, b}, {a, c}, {b, c}, {c, d}} ?

Here V = {a, b, c, d} are the vertices and E = {{a,b}, {a,c}, {b,c}, {c,d}} are the edges of the graph.

Degree of a Vertex

The degree of a vertex V of a graph G (denoted by deg(V)) is the number of edges incident with (connected to) that vertex.

For the graph above, the degree of each vertex is −

Even and Odd Vertex

If the degree of a vertex is even, it is called an even vertex. If the degree of a vertex is odd, it is called an odd vertex. In the graph above, vertices a and b are even vertices (degree 2), while c and d are odd vertices (degrees 3 and 1).

Degree of a Graph

The degree of a graph is the largest vertex degree in that graph. For the above graph, the degree of the graph is 3 (the degree of vertex c).

The Handshaking Lemma

In any graph, the sum of the degrees of all vertices is equal to twice the number of edges −

∑ deg(V_i) = 2 × |E|

For the above graph −

Sum of degrees = deg(a) + deg(b) + deg(c) + deg(d)
               = 2 + 2 + 3 + 1
               = 8

Number of edges = |E| = 4

2 × |E| = 2 × 4 = 8  ?

The sum of all degrees (8) equals twice the number of edges (2 × 4 = 8), confirming the Handshaking Lemma.

Conclusion

Vertices are the points and edges are the connections in a graph. The degree of a vertex counts its incident edges, and the Handshaking Lemma guarantees that the sum of all vertex degrees always equals twice the total number of edges.