# Adjacency matrix on simple graphs

It is known that one of the eigenvalues in the $k$-regular graph is $k$.

I have to prove that for a connected graph with eigenvalue $\Delta$, in which $\Delta$ is the maximum degree in G, the graph is regular.

For a generic graph, all it's eigenvalues are less or equal than $\Delta$, mi question is thus the following, is there a graph which has eigenvalue $\Delta$ and it's not regular? If not, then the statement is easily to prove.