Extremal Graph Theory for Degree Sequences

TL;DR

This survey explores extremal problems in graph theory focusing on graphs with fixed degree sequences, analyzing maximum or minimum spectral and structural invariants across various graph classes.

Contribution

It compiles recent results on extremal graphs with fixed degree sequences, characterizing those with extremal spectral and structural properties, and discusses open conjectures.

Findings

Characterization of extremal graphs for spectral radii

Identification of extremal graphs for graph invariants like Wiener and Harary indices

Formulation of conjectures on extremal properties in degree sequence sets

Abstract

This paper surveys some recent results and progress on the extremal prob- lems in a given set consisting of all simple connected graphs with the same graphic degree sequence. In particular, we study and characterize the extremal graphs having the maximum (or minimum) values of graph invariants such as (Laplacian, p-Laplacian, signless Laplacian) spectral radius, the first Dirichlet eigenvalue, the Wiener index, the Harary index, the number of subtrees and the chromatic number etc, in given sets with the same tree, unicyclic, graphic degree sequences. Moreover, some conjectures are included.