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

Abstract

Suppose π = (d 1, d 2, …, d n) and π′ = (d′ 1, d′ 2, …, d′ n) are two positive non- increasing degree sequences, write π ⊳ π′ if and only if π ≠ π′, (Equation presented)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) < <inf>ρ (G′) (respectively, μ (G) < <inf>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.