# Premagic and Ideal Flow Matrices

## TL;DR

This paper explores properties of premagic matrices, their applications to strongly connected directed graphs, and discusses relationships with Markov chains, ideal flow, and random walks, providing theoretical insights into flow conservation and network analysis.

## Contribution

It introduces and analyzes properties of premagic matrices, connecting them to network flow, Markov chains, and random walks, offering new theoretical insights.

## Key findings

- Premagic matrices have specific properties related to flow conservation.
- Connections between premagic matrices, Markov chains, and random walks are established.
- The paper provides proofs of properties of premagic matrices and their applications in network analysis.

## Abstract

Several interesting properties of a special type of matrix that has a row sum equal to the column sum are shown with the proofs. Premagic matrix can be applied to strongly connected directed network graph due to its nodes conservation flow. Relationships between Markov Chain, ideal flow and random walk on directed graph are also discussed.