Define the terms 'degree' and 'order' in Graph Theory.

In Graph Theory, the terms 'degree' and 'order' are used to describe certain properties of vertices in a graph.

1. Degree:
The degree of a vertex in a graph refers to the number of edges incident to that vertex. In other words, it represents the number of connections or links a vertex has with other vertices in the graph. The degree of a vertex is denoted by 'deg(v)', where 'v' is the vertex.

There are two types of degrees in graph theory:
- In-degree: In a directed graph, the in-degree of a vertex is the number of edges pointing towards that vertex.
- Out-degree: In a directed graph, the out-degree of a vertex is the number of edges originating from that vertex.

The degree of a vertex is a fundamental concept in graph theory as it provides important information about the connectivity and structure of the graph. For example, in a simple undirected graph, the sum of degrees of all vertices is equal to twice the number of edges in the graph (Handshaking Lemma).

2. Order:
The order of a graph refers to the total number of vertices present in the graph. It is denoted by '|V|', where 'V' represents the set of vertices in the graph. In simple terms, the order of a graph gives us the count of vertices in the graph.

The order of a graph is an essential characteristic that helps in understanding the size and complexity of the graph. It provides a basic understanding of the number of elements or entities present in the graph.

To summarize, the degree of a vertex in a graph represents the number of edges incident to that vertex, while the order of a graph refers to the total number of vertices present in the graph. Both these terms play a crucial role in analyzing and understanding the properties and structure of graphs in Graph Theory.