Some results between the largest eigenvalue and the maximum degree of graphs

Let G be a graph of order n with eigenvalues. The λ1(G) ≥ ... ≥ λn (G) largest eigenvalue of G, λ1(G) , is called the spectral radius of G. Let Δ(G) be the maximum degree of the vertices of G and β(G) = Δ(G)- λ1(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 among all non-regular connected graphs G. We obtain that for every tree T with n ≥ 3 vertices, β(Sn)≥ β(T)≥β(Pn). Moreover, we prove that in the right side the equality holds if and only if T T≈Pn and in the other side the equality holds if and only if T≈Sn, where Pn, Sn are the path and the star on n vertices, respectively