The majorization theorem of extremal pseudographs

A pseudograph is a graph in which both loops and multiple edges are permitted. Suppose π=(d1,d2,...,dn)π=(d1,d2,...,dn) and π′=(d1′,d2′,...,dn′) are two positive non-increasing degree sequences, we write π◁π′π◁π′ if and only if π≠π′π≠π′, ∑i=1ndi=∑i=1ndi′, and ∑i=1jdi≤∑i=1jdi′ for all j=1,2,...,nj=1,2,...,n. Let Γ(π)Γ(π) be the class of connected undirected pseudographs with degree sequence π  . Let ρ(G)ρ(G) and μ(G)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)Γ(π) are characterized. Furthermore, we show that if π◁π′π◁π′, G   and G′G′ are the pseudographs with the largest (respectively, signless Laplacian) spectral radii in Γ(π)Γ(π) and Γ(π′)Γ(π′), respectively, then ρ(G)<ρ(G′)ρ(G)<ρ(G′) and μ(G)<μ(G′)μ(G)<μ(G′).