Answer:
The number of branches incident at a node in a graph is called the degree of the node. It is a fundamental concept in graph theory and is used to describe the connectivity of a node in a graph. The degree of a node is defined as the number of edges that are connected to that node.

Explanation:
In a graph, nodes represent entities, and edges represent relationships or connections between these entities. The degree of a node is a measure of how many connections or branches are incident at that node.

Example:
Consider a simple graph with three nodes A, B, and C. If there are two edges connecting node A to node B and one edge connecting node A to node C, then the degree of node A would be 3. This means that there are three branches or edges incident at node A.

Significance:
The degree of a node provides important information about the connectivity of a graph. It can help determine the complexity of a network, identify important nodes or hubs, and analyze the flow of information or resources in a system.

Properties:
- In an undirected graph, the degree of a node is equal to the number of edges connected to that node.
- In a directed graph, the degree of a node is the sum of the in-degree (number of edges coming into the node) and the out-degree (number of edges going out from the node).
- The minimum degree of a node in a graph is 0, indicating that it is not connected to any other node.
- The maximum degree of a node in a graph is the number of nodes minus one, indicating that it is connected to all other nodes in the graph.

Conclusion:
In graph theory, the degree of a node is a fundamental concept that describes the connectivity of a node in a graph. It is defined as the number of branches or edges incident at that node. The degree of a node provides important information about the structure and connectivity of a graph and is used in various applications such as network analysis, social network analysis, and routing algorithms.