On the subdirect product of graph bundles

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.