The Majorization Theorems of Single-Cone Trees and Single-Cone Unicyclic Graphs

A single-cone tree (unicyclic graph) is the join of a complete graph K 1 and a tree (unicyclic graph). Suppose π = ( d 1 , d 2 , … , d n ) and π ′ = ( d 1 ′ , d 2 ′ , … , d n ′ ) are 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 - 1 . We use J π to denote the class of single-cone trees (unicyclic graphs) with degree sequence π . Suppose that π and π ′ are two different non-increasing degree sequences of single-cone trees (unicyclic graphs). Let ρ and ρ ′ be the largest spectral radius of the graphs in J π and J π ′ , respectively, μ and μ ′ be the largest signless Laplacian spectral radius of the graphs in J π and J π ′ , respectively. In this paper, we prove that if π ⊲ π ′ , then ρ < ρ ′ and μ < μ ′ .