# Enumeration on graph mosaics

## TL;DR

This paper extends the concept of knot mosaics to graph mosaics, developing an enumeration algorithm for counting all possible graph mosaics using an extended state matrix approach.

## Contribution

It introduces a graph mosaic system with 16 tiles and an enumeration algorithm, expanding previous knot mosaic enumeration methods to graphs with vertices of valence 3 and 4.

## Key findings

- Developed an exact enumeration algorithm for graph mosaics.
- Extended the state matrix method to graph mosaics.
- Provided a framework for counting graph diagrams with vertices of valence 3 and 4.

## Abstract

Since the Jones polynomial was discovered, the connection between knot theory and quantum physics has been of great interest. Lomonaco and Kauffman introduced the knot mosaic system to give a definition of the quantum knot system that is intended to represent an actual physical quantum system. Recently the authors developed an algorithm producing the exact enumeration of knot mosaics, which uses a recursion formula of state matrices. As a sequel to this research program, we similarly define the (embedded) graph mosaic system by using sixteen graph mosaic tiles, representing graph diagrams with vertices of valence 3 and 4. And we extend the algorithm to produce the exact number of all graph mosaics. The magnified state matrix that is an extension of the state matrix is mainly used.