The majorization theorem of connected graphs

Abstract

Let π = ( d 1 , d 2 , … , d n ) and π ′ = ( d 1 ′ , d 2 ′ , … , d n ′ ) be two non-increasing degree sequences. We say π is majorizated by π ′ , denoted by π ◁ π ′ , if and only if π ≠ π ′ , ∑ i = 1 n d i = ∑ i = 1 n d i ′ , and ∑ i = 1 j d i ⩽ ∑ i = 1 j 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 π 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 C π and C π ′ , 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 ′ ) .