Out Degree Sequence And In Degree Sequence

Out-Degrees and In-Degrees of a Vertex

Definition: For a directed graph $G = (V(G), E(G))$ and a vertex $x_1 \in V(G)$, the Out-Degree of $x_1$ refers to the number of arcs incident from $x_1$. That is, the number of arcs directed away from the vertex $x_1$. The In-Degree of $x_1$ refers to the number of arcs incident to $x_1$. That is, the number of arcs directed towards the vertex $x_1$.

We denote the out-degree of a vertex $x_1$ by the notation $\mathrm{outdeg}(x_1)$ while we denote the in-degree of the same vertex by the notation $\mathrm{indeg}(x_1)$.

For example, let's look at the following directed graph.

In the following graph above, the out-degrees of each vertex are in blue, while the in-degrees of each vertex are in red.

Out-Degree Sequence and In-Degree Sequence of a Graph

Definition: For a directed graph $G = (V(G), E(G))$, the Out-Degree Sequence is a sequence obtained by ordering the out-degrees of all vertices in $V(G)$ in increasing order. The In-Degree Sequence is a sequence obtained by ordering the in-degrees of all vertices in $V(G)$ in increasing order.

From the graph earlier, the out-degree sequence (blue degrees) is $(0, 1, 1, 1, 2, 3)$, while the in-degree sequence (red degrees) is $(0, 1, 1, 2, 2, 2)$.

Remark: The degree sequence of a graph $G$ that is NOT directed has the degrees of each vertex ordered in decreasing order, while the degree sequence of a graph $G$ that is directed has the degrees of each vertex ordered in increasing order.