Regular graph in the context of Indegree

In graph theory, the hypercube graph ${\displaystyle Q_{n}}$ is the edge graph of the ${\displaystyle n}$-dimensional hypercube, that is, it is the graph formed from the vertices and edges of the hypercube. For instance, the cube graph ${\displaystyle Q_{3}}$ is the graph formed by the 8 vertices and 12 edges of a three-dimensional cube.${\displaystyle Q_{n}}$ has ${\displaystyle 2^{n}}$ vertices, ${\displaystyle 2^{n-1}n}$ edges, and is a regular graph with ${\displaystyle n}$ edges touching each vertex.

The hypercube graph ${\displaystyle Q_{n}}$ may also be constructed by creating a vertex for each subset of an ${\displaystyle n}$-element set, with two vertices adjacent when their subsets differ in a single element, or by creating a vertex for each ${\displaystyle n}$-digit binary number, with two vertices adjacent when their binary representations differ in a single digit. It is the ${\displaystyle n}$-fold Cartesian product of the two-vertex complete graph, and may be decomposed into two copies of ${\displaystyle Q_{n-1}}$ connected to each other by a perfect matching.

In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex; in a multigraph, a loop contributes 2 to a vertex's degree, for the two ends of the edge. The degree of a vertex ${\displaystyle v}$ is denoted ${\displaystyle \deg(v)}$ or ${\displaystyle \deg v}$. The maximum degree of a graph ${\displaystyle G}$ is denoted by ${\displaystyle \Delta (G)}$, and is the maximum of ${\displaystyle G}$'s vertices' degrees. The minimum degree of a graph is denoted by ${\displaystyle \delta (G)}$, and is the minimum of ${\displaystyle G}$'s vertices' degrees. In the multigraph shown on the right, the maximum degree is 5 and the minimum degree is 0.

In a regular graph, every vertex has the same degree, and so we can speak of the degree of the graph. A complete graph (denoted ${\displaystyle K_{n}}$, where ${\displaystyle n}$ is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum possible degree, ${\displaystyle n-1}$.