Prove that vector 1 (vector with all entries as 1) is an eigenvector of the adjacency matrix corresponding to eigenvalue k if and only if the graph is a k-regular graph.

What will be an ideal response?

Proof. For a

k-regular graph, the adjacency matxix contains exacly
k number of 1’s, and as a result sum of each row is
k. Therefore,

A1=k1.

Hence,1 is an eigenvector corresponding to the eigenvalue k. Now we need to prove the reverse statement. Let us assume that
1 is the eigenvector corresponding to the eigenvalue k. We have to prove that it is possible for only k-regular graphs.
Based on the assumption, we have

A1=k1

Therefore, the sum of each row of the adjacency matrix is
k. This is possible only for k-regular graphs.