Are all adjacency matrices of connected graph diagonalizable?

Question

The related question has been asked in the following link:
Are all adjacency matrices (graph theory) diagonalizables?

However, the answer does not give a formal proof or explicit answer.

My question is:

For any connected graph (undirected graph), is its adjacency matrix diagonalizable?

Note: it is obviously that it does not have to be full rank, for example a tree graph.

Answer 1

The answer is yes for undirected graphs, since the adjacency matrix of a finite undirected graph (not necessarily connected) is a symmetric real matrix.