Suppose Ï = (d_1,d_2,...,d_n) and Ïâ² = (dâ²_1,dâ²_2,...,dâ²_n) are two positive non- increasing degree sequences, write Ï â³ Ïâ² if and only if Ï \neq Ïâ², \sum_{i=1}^n d_i = \sum_{i=1}^n dâ²_i, and \sum_{i=1}^j d_i ⤠\sum_{i=1}^j dâ²_i for all j = 1, 2, . . . , n. Let Ï(G) and μ(G) be the spectral radius and signless Laplacian spectral radius of G, respectively. Also let G and Gâ² be the extremal graphs with the maximal (signless Laplacian) spectral radii in the class of connected graphs with Ï and Ïâ² as their degree sequences, respectively. If Ï â³ Ïâ² can deduce that Ï(G) < Ï(Gâ²) (respectively, μ(G) < μ(Gâ²)), then it is said that the spectral radii (respectively, signless Laplacian spectral radii) of G and Gâ² satisfy the majorization theorem. This paper presents a survey to the recent results on the theory and application of the majorization theorem in graph spectrum and topological index theory.
The signless Laplacian spectral radius of bicyclic graphs with prescribed degree sequences