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

Suppose π = (d1, d2, . . . , dn) and π = (d1, d ′ 2 , . . . , dn) are two positive nonincreasing degree sequences, write π ⊳ π if and only if π 6= π, ∑n i=1 di = ∑n i=1 d ′ i, and ∑j i=1 di ≤ ∑j i=1 di for all j = 1, 2, . . . , n. Let ρ(G) and μ(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 π and π as their degree sequences, respectively. If π ⊳ π can deduce that ρ(G) < ρ(G) (respectively, μ(G) < μ(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.