A further result on majorization theorem

Abstract

Let π = (d 1, d 2, … , d n ) and be two non-increasing degree sequences. We say π is majorizated by π′, denoted by π ◃ π′, if and only if π ≠ π′, and for all j = 1, 2, … , n − 1. We use C π to denote the class of connected graphs with degree sequence π. Let ρ(G) be the spectral radius, i.e. the largest eigenvalue of the adjacent matrix of G. In this article, we prove that if π ◃ π′, B and B′ are the bicyclic graphs with the greatest spectral radius in C π and , respectively, then ρ(B) < ρ(B′). And we give an example to show that this majorization theorem is not true for tricyclic graphs.