### Abstract:

Let G be a graph with the eigenvalues λ1(G) >· · · > λn(G). The largest eigenvalue of G, λ1(G), is called the spectral radius of G. Let β(G) = ∆(G) − λ1(G), where ∆(G) is the maximum degree of vertices of G. It is known that if G is a connected graph, then β(G) > 0 and the equality holds if and only if G is regular. In this paper we study the maximum value and the minimum value of β(G) among all non-regular connected graphs. In particular we show that for every tree T with n > 3 vertices, n − 1 − √ n − 1 > β(T) > 4 sin2 (π2n+2 ). Moreover, we prove that in the right side the equality holds if and only if T ∼= Pn and in the other side the equality holds if and only if T ∼= Sn, where Pn and Sn are the path and the star on n vertices, respectively.