**abstract:**
Magnitude was first introduced by Leinster in 2008 1. It is a notion

analogous to the Euler characteristic of a category, and it captures the struc- ture and complexity of a metric space. Magnitude homology was defined in

2014 by Hepworth and Willerton 2 as a categorification of magnitude in the context of simple undirected graphs, and although the construction of the boundary map suggests that magnitude homology groups are strongly influenced by the graph substructures, it is not straightforward to detect such subgraphs. In this talk, I introduce eulerian magnitude homology 3.

I will do this by defining the eulerian magnitude chain complex, a subcom- plex of the magnitude chain complex exhibiting a more explicit connection

to the combinatorics of the graph. I will illustrate how eulerian magnitude homology enables a more accurate analysis of graph substructures and then apply these results to Erd ̋os-R ́enyi random graphs and obtain an asymptotic estimate for the Betti numbers of the eulerian magnitude homology groups on the diagonal. Finally, I will discuss the regimes where an Erd ̈os-R ́enyi random graph has torsion free eulerian magnitude homology groups 4. References 1 T. Leinster,The Euler characteristic of a category, Documenta Mathematica (13) (2008) 21–49.

2 R. Hepworth, S. Willerton, Categorifying the magnitude of a graph. Homo- topy, Homology and Applications 16(2) (2014) 1–30.

3 C. Giusti, G. Menara, Eulerian magnitude homology: subgraph structure and random graphs. arXiv: 2403.09248 (2024).

4 G. Menara, On torsion in eulerian magnitude homology of Erdos-Renyi ran- dom graphs. arXiv: 2409.03472 (2024).

Fri 4 Oct, 9:30 - 10:30, Aula Dini

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