Largest eigenvalue of the adjacency matrix of a graph

Question

Let $G$ be a connected graph with $n$ vertices and $m$ edges. Assume that $\lambda_1$ is the largest eigenvalue of adjacency matrix of $G$. I know that $\lambda_1\geq 2m/n$ with equality holding if and only if $G$ is a regular graph, but I cannot prove it. Could you please help me to prove it?

Answer 1

HINT:

Let $A$ be the adjacency matrix matrix of the graph, $\lambda_1$ its largest eigenvalue. Then $\lambda_1 I - A$ is positive semi-definite. Now the sum of entries of a positive semi-definite matrix is $\ge 0$. We conclude that $$n \lambda_1 - 2 m \ge 0$$

Note that the sum of entries of $\lambda_1 I - A$ equals $$\langle (\lambda_1 I - A) v , v\rangle$$ where $v = (1,\ldots, 1)$. If we have equality, $v$ must be in the kernel of $\lambda_1 I - A$.