Graph algebras

TL;DR

This paper establishes the optimal bound on the number of paths of a given length in certain graphs and introduces algorithms to transform graphs to achieve this bound, with implications for graph algebras.

Contribution

It provides the first proof of the optimal path count bound in graphs with no loops using terminating algorithms that reshape graphs without reducing path counts.

Findings

Derived the optimal bound for path counts in loopless graphs.

Developed algorithms to transform graphs to reach the optimal bound.

Connected graph algebra structures to C*-algebra completions.

Abstract

This introduction to graphs and graph algebras provides the optimal bound for the number of all paths of length $k$ in a graph with $N\geq k$ edges and no loops. Our proof relies on a construction of a number of terminating algorithms that reshape such graphs without ever decreasing the number of paths of length $k$. The key two algorithms work in turns each of them ending with a graph to which the other algorithm can be applied. Finally, one arrives at a specific graph realizing the optimal bound. Herein graph algebras mean path algebras and Leavitt path algebras. For the ground field $\mathbb{C}$ of complex numbers, the latter are viewed as dense subalgebras in their universal C*-completions called graph C*-algebras.