In an undirected graph, the number of links of vertex $i$ is called the degree $k_{i}$ of the vertex, and can be expressed in terms of the adjacency matrix as
$$k_{i}=\sum_{i=1}^{N} a_{i j}$$
In a directed graph, it is possible to define the in-degree $k_{i}^{i n}$ and the outdegree $k_{i}^{\text {out }}$ as the number of in-coming and out-going links of vertex $i$ respectively, which can be expressed as
$$k_{i}^{i n}=\sum_{i=1}^{N} a_{j i} \quad k_{i}^{\text {out }}=\sum_{i=1}^{N} a_{i j}$$

The total number of links $L$ in a directed graph is given by
$$L=\sum_{i=1}^{N} k_{i}^{i n}=\sum_{i=1}^{N} k_{i}^{\text {out }}=\sum_{i, j} a_{i j}$$
while in an undirected graph it is given by
$$L=\frac{1}{2} \sum_{i=1}^{N} k_{i}=\frac{1}{2} \sum_{i, j} a_{i j}=\sum_{i<j} a_{i j}$$
since each pair of vertices must be counted only once.

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