I am self-studying graph (spectral) theory and trying to prove the following problem. Assume we have a graph $ G = (V,E)$ with adjacency matrix $ A$ and diagonal degree matrix $ D$ (where each diagonal entry represents the degree of the corresponding node). Define the Laplacian matrix $ L = D-A$ . Suppose that the number of non-isolated vertices in this graph is $ \tilde{n}$ , then the claim is that the largest eigenvalue of $ L$ , $ \lambda_1$ , must satisfy $ $ \lambda_1 \leq \tilde{n}$ $

I am not sure how to show this. The trace of $ L$ , equals the sum of degrees of the graph, also equals the sum of all eigenvalues. The algebraic multiplicity of the eigenvalue zero indicates the number of connected components, but I am not sure how to precede from here.

Any thought is appreciated.