The majorization theorem of connected graphs

Let π=(d1,d2,…,dn) and be two non-increasing degree sequences. We say π is majorizated by π′, denoted by π◁π′, if and only if , and for all j=1,2,…,n. If the degree of vertex v is (resp. not) equal to 1, then we call v a pendant (resp. non-pendant) vertex of G. We use Cπ to denote the class of connected graphs with degree sequence π. Suppose π and π′ are two non-increasing c-cyclic degree sequences. Let G and G′ be the graphs with greatest spectral radii in Cπ and Cπ′, respectively. In this paper, we shall prove that if π◁π′,G and G′ have the same number of pendant vertices, and the degrees of all non-pendant vertices of G′ are greater than c, then ρ(G)<ρ(G′).