Recent results on the majorization theory of graph spectrum and topological index theory

Abstract

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.