On the subdirect product of graph bundles

TL;DR

This paper introduces a novel graph bundle operation to model the subdirect product of finite groups at the graph level, enabling a new perspective on graph products and related algebraic structures.

Contribution

It defines a graph bundle-based operation that captures the subdirect product of groups through Cayley graph constructions, linking group theory and graph theory.

Findings

Cayley graph of subdirect product described as total space of graph bundle product

Introduction of network K-theory group of a graph inspired by topological K-theory

Establishment of a functor from graphs to abelian groups

Abstract

The subdirect product of two finite groups $A$ and $B$ is defined as a subgroup of the direct product $A \times B$, which is a well-known notion in finite group theory. While it is clear that, under appropriate choices of sets of generators $S$, $S_A$ and $S_B$, the Cayley graph $Cay(A \times B, S)$ corresponds to the Cartesian product $Cay(A, S_A) \square Cay(B, S_B)$ of two graphs, there is no analogue at the level of graph product that reflects the notion of subdirect product of groups. This is precisely the problem which we discuss here. By using the concept of graph bundles and the corresponding pullbacks, we introduce an operation on graph bundles such that the Cayley graph of the subdirect product of two groups can be described as the total space of the product of the Cayley graphs. This allows us to define the so-called ``network $K$-theory group of a graph'', inspired by the notion of topological $K$-theory, and we are able to investigate an interesting functor from the category of graphs to the category of abelian groups.