A further result on majorization theorem

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) di = 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 - 1. We use C(pi) to denote the class of connected graphs with degree sequence pi. Let rho(G) be the spectral radius, i. e. the largest eigenvalue of the adjacent matrix of G. In this article, we prove that if pi (sic) pi', B and B' are the bicyclic graphs with the greatest spectral radius in C(pi) and C(pi'), respectively, then rho(B) < rho(B'). And we give an example to show that this majorization theorem is not true for tricyclic graphs.