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

Suppose I€ = (d_1,d_2,...,d_n) and I€â€² = (d′_1,d′_2,...,d′_n) are two positive non- increasing degree sequences, write I€ ⊳ I€â€² if and only if I€ \neq I€â€², \sum_{i=1}^n d_i = \sum_{i=1}^n d′_i, and \sum_{i=1}^j d_i ≤ \sum_{i=1}^j d′_i for all j = 1, 2, . . . , n. Let I(G) and I¼(G) be the spectral radius and signless Laplacian spectral radius of G, respectively. Also let G and G′ be the extremal graphs with the maximal (signless Laplacian) spectral radii in the class of connected graphs with I€ and I€â€² as their degree sequences, respectively. If I€ ⊳ I€â€² can deduce that I(G) < I(G′) (respectively, I¼(G) < I¼(G′)), then it is said that the spectral radii (respectively, signless Laplacian spectral radii) of G and G′ 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.