What is the relationship between the sum of degrees of all vertices in a network and the number of edges?

The sum of the degrees of all vertices in a network (or a graph) is intrinsically linked to the number of edges present in that graph. Each edge connects two vertices, and therefore contributes to the degree count of both of those vertices.

When you take into account this connection, it becomes clear that each edge effectively contributes two to the total degree count (one for each vertex it connects). Consequently, if you have a certain number of edges in the graph, multiplying that number by two will yield the total sum of the degrees of all vertices.

For example, if there are three edges in the network, they would connect several vertices, adding to the degree sums of those involved vertices. Since each edge counts as contributing to the degrees of two vertices, the total sum of the vertex degrees would be six in this case, which corresponds to two times the number of the edges.

Understanding this relationship is fundamental in graph theory and plays a critical role in various mathematical applications, such as determining the connectivity and structure of networks.