Bounding the largest eigenvalue of trees in terms of the largest vertex degree (Q1863520)

Let \(\lambda_1(G)\) denote the largest eigenvalue of the adjacency matrix and let \(\mu_1(G)\) denote the largest eigenvalue of the Laplacian matrix \(L(G)= QQ^t\) of a graph \(G\) (\(Q\) is an incidence matrix of \(G\)). The author shows that if a tree \(T\) has the largest vertex degree \(\Delta\) then \(\lambda_1(T)< 2\sqrt{\Delta- 1}\) and \(\mu_1(T)< \Delta+ 2\sqrt{\Delta-1}\).