The majorization theorem of connected graphs

Let pi = (d(1), d(2), ... , d(n)) and pi' = (d(1)', d(2)', ... , d(n)') be two non-increasing degree sequences. We say pi is majorizated by pi', denoted by pi (sic) pi', if and only if pi not equal pi', Sigma(n)(i=1) d(i) = Sigma(n)(i=1) d(i)', and Sigma(j)(i=1), d(i) <= Sigma(j)(i=1), d(i)' 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 C-pi to denote the class of connected graphs with degree sequence pi. Suppose pi and pi' are two non-increasing c-cyclic degree sequences. Let G and G' be the graphs with greatest spectral radii in C-pi and C-pi', respectively. In this paper, we shall prove that if pi (sic) pi', 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 rho(G) < rho(G'). (c) 2009 Elsevier Inc. All rights reserved.