Arbitrarily regularizable graphs

TL;DR

This paper introduces the concept of arbitrarily regularizable graphs, exploring conditions for their regularization, analyzing the computational complexity, and formulating the problem as a linear programming model.

Contribution

It extends the study of regularizable graphs by defining arbitrarily regularizable graphs and providing necessary and sufficient topological conditions, along with a linear programming formulation.

Findings

Characterized conditions for arbitrary regularizability based on graph topology

Proved the problem's computational complexity and formulated it as a linear program

Extended existing literature on positive and nonnegative regularizable graphs

Abstract

A graph is regularizable if it is possible to assign weights to its edges so that all nodes have the same degree. Weights can be positive, nonnegative or arbitrary as soon as the regularization degree is not null. Positive and nonnegative regularizable graphs have been thoroughly investigated in the literature. In this work, we propose and study arbitrarily regularizable graphs. In particular, we investigate necessary and sufficient regularization conditions on the topology of the graph and of the corresponding adjacency matrix. Moreover, we study the computational complexity of the regularization problem and characterize it as a linear programming model.