If is the adjacency matrix of a -regular graph and , every row sum of is , soIf the graph is connected, the -eigenspace is exactly .
A connected regular graph has exactly two distinct adjacency eigenvalues if and only if it is a complete graph with . Its eigenvalues are on the constant vectors and on their orthogonal complement.
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A **regular graph** is a type of graph in which every vertex has the same number of edges. This common degree is known as the **degree** of the regular graph. There are two main types of regular graphs: 1. **k-regular**: A graph is k-regular if every vertex has exactly k edges. For example: - A 1-regular graph consists of disjoint edges (pairs of vertices).