The majorization theorem of extremal pseudographs

Abstract A pseudograph is a graph in which both loops and multiple edges are permitted. Suppose π = ( d 1 , d 2 , . . . , d n ) and π ′ = ( d 1 ′ , d 2 ′ , . . . , d n ′ ) are two positive non-increasing degree sequences, we write 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 . Let Γ ( π ) be the class of connected undirected pseudographs with degree sequence π . Let ρ ( G ) and μ ( G ) be the spectral radius and signless Laplacian spectral radius of G , respectively. In this paper, the extremal pseudographs with the largest (respectively, signless Laplacian) spectral radii in Γ ( π ) are characterized. Furthermore, we show that if π ◁ π ′ , G and G ′ are the pseudographs with the largest (respectively, signless Laplacian) spectral radii in Γ ( π ) and Γ ( π ′ ) , respectively, then ρ ( G ) ρ ( G ′ ) and μ ( G ) μ ( G ′ ) .