Algebraic Properties
Let A be the adjacency matrix of a graph. Then the graph is regular if and only if is an eigenvector of A. Its eigenvalue will be the constant degree of the graph. Eigenvectors corresponding to other eigenvalues are orthogonal to, so for such eigenvectors, we have .
A regular graph of degree k is connected if and only if the eigenvalue k has multiplicity one.
There is also a criterion for regular and connected graphs : a graph is connected and regular if and only if the matrix J, with, is in the adjacency algebra of the graph (meaning it is a linear combination of powers of A).
Let G be a k-regular graph with diameter D and eigenvalues of adjacency matrix . If G is not bipartite
where .
Read more about this topic: Regular Graph
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