Distance ideals of digraphs

We focus on strongly connected, strong for short, digraphs since in this setting distance is defined for every pair of vertices. Distance ideals generalize the spectrum and Smith normal form of several distance matrices associated with strong digraphs. We introduce the concept of pattern which allow us to characterize the family Γ1 of digraphs with only one trivial distance ideal over Z. This result generalizes an analogous result for undirected graphs that states that connected graphs with one trivial ideal over Z consists of either complete graphs or complete bipartite graphs. It turns out that the strong digraphs in Γ1 consists in the circuit with 3 vertices and a family Λ of strong digraphs that contains complete graphs and complete bipartite graphs, regarded as digraphs. We also compute all distance ideals of some strong digraphs in the family Λ. Then, we explore the distance ideals of circuits, which turns out to be an infinite family of digraphs with unbounded diameter in Γ2, that is, digraphs with two trivial distance ideals.

Recommended citation: Carlos A. Alfaro, Teresa I. Hoekstra-Mendoza, Juan Pablo Serrano and Ralihe R. Villagrán. "Distance ideals of digraphs." Applied Mathematics and Computation. 500, (2025), 129430. https://doi.org/10.1016/j.amc.2025.129430