In this paper, we introduce a new method to compute the magnitude homology of general graphs. To each direct sum component of the magnitude chain complexes, we assign a pair of simplicial complexes whose simplicial chain complex is isomorphic to it. First we state our main theorem specialized to trees, which gives another proof for the known fact that trees are diagonal. After that, we consider general graphs, which may have cycles. We also demonstrate some computation as an application.

more »

... t In this paper, we introduce a new method to compute the magnitude homology of general graphs. To each direct sum component of the magnitude chain complexes, we assign a pair of simplicial complexes whose simplicial chain complex is isomorphic to it. First we state our main theorem specialized to trees, which gives another proof for the known fact that trees are diagonal. After that, we consider general graphs, which may have cycles. We also demonstrate some computation as an application. Abstract In this paper, we introduce a new method to compute the magnitude homology of general graphs. To each direct sum component of the magnitude chain complexes, we assign a pair of simplicial complexes whose simplicial chain complex is isomorphic to it. First we state our main theorem specialized to trees, which gives another proof for the known fact that trees are diagonal. After that, we consider general graphs, which may have cycles. We also demonstrate some computation as an application. YASUHIKO ASAO and KENGO IZUMIHARA Theorem 1.1. Let a, b be vertices of a graph G, and fix an integer 3. Then we can construct a pair of simplicial complexes (K (a, b), K (a, b) ) which satisfies C * (K (a, b), K (a, b) where s denotes suspension of the complex detailed in section 2. In particular, we have for k, 3. Moreover, for k = 2, we also have whereH * denotes the reduced homology group. Our theorem yields an interpretation of magnitude homology groups as homology groups of a simplicial complex. Therefore, our method allows us to apply sophisticated tools of homotopy theory. In the special cases of 2 4 we obtain a visualization of the magnitude chain complex since the dimensions of the corresponding simplical complexes are 0, 1, and 2, respectively. The organization of this paper is the following: After giving some basic definitions and notation in section 2, we first give a new method for computing the magnitude homology of trees based on our simplicial strategy. The computational results thus obtained coincide with [3, Corollary 31]. The computation for a tree is simpler than that for the general graphs studied in the following section, because of the fact that the magnitude chain complex of a tree can be decomposed into simple ones. In section 4, we give a proof of our main theorem, and compute the magnitude homology of the graph Sq 2 introduced in [7] as an application. Preliminaries In this section, we recall some basic definitions for graphs and their magnitude homology together with related notation. The main definitions are taken from [3]. Simplicial complexes Definition 2.1. Let V be a set, and let P (V ) be its power set. A subset S ⊂ P (V ) \ {∅} is called a simplicial complex if it satisfies that B ∈ S for every ∅ = B ⊂ A ∈ S. Proposition 3.2. Let G be a tree. We have the following direct sum decomposition MC * , (a, b) = x∈P (a,b) MC * , (x), for each a, b ∈ V (G) and 1.