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

Suppose π = (d₁, d₂, …, 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) < _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.