چكيده فارسي :
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
