How is the degree of a vertex defined?

The degree of a vertex in a graph is defined as the number of edges that are incident to it, which means it counts how many edges are connected to that specific vertex. This concept is fundamental in graph theory because it helps to understand the connectivity and relationships between different vertices in a graph.

In simple terms, if a vertex is connected to three edges, its degree is 3. This measure is especially useful when analyzing the properties of the graph, like determining whether a graph is connected, finding potential paths, and identifying important vertices such as hubs in network theory.

Other choices describe aspects not related to the specific definition of a vertex's degree. For example, counting the number of vertices doesn't provide information about a particular vertex's connections. Similarly, the total length of the edges or the sum of all vertex values misattributes the concept to characteristics of the edges or the vertices themselves rather than their connectivity. Hence, the definition that best aligns with the established concept in graph theory is that the degree of a vertex is determined by the number of edges connected to it.