Let G is a connected undirected graph and G\e is a graph obtained by removing a random link e from the graph G. Let $\lambda_1(A(G))$ is the largest eigenvalue of the adjacency matrix of graph G. Is $\lambda_1(A(G))-\lambda_1(A(G\e)) \geq 0$? in other words, deleting a link will never increase the largest eigenvalue of the resulting graph. I think it is true but still cannot prove it.