The Lorenz order in graph theory: A new proof and extension of the theorems of Hakimi and of Havel-Hakimi

TL;DR

This paper introduces a new proof and extension of classical theorems in graph theory using Lorenz majorization, linking degree sequence dominance to graphical realizability and generalizing results for connected networks.

Contribution

It provides a simple proof and a generalization of the Havel-Hakimi theorem using Lorenz order, extending classical results to broader classes of networks.

Findings

Lorenz order determines graphical realizability of degree sequences.

A generalized Havel-Hakimi theorem using Lorenz majorization.

Characterization of c-graphical sequences via Lorenz order.

Abstract

This paper studies the relation between the Lorenz majorization order and the realizability of degree sequences X of a network in the sense of being graphical or connected graphical (c-graphical) or not. We prove the main result that, if X is dominated (in the Lorenz majorization sense) by X' and X' is (c-) graphical, the X is also (c-) graphical. We present a simple proof and a generalization of the Havel-Hakimi theorem, using the Lorenz order formalism. From this, a classical result of Hakimi on trees follows but also a new generalization to general connected networks. From this, a characterization of c-graphical sequences in terms of the Lorenz majorization order is given.