In this article, we show that magnitude homology and path homology are closely related, and we give some applications. We define differentials MH 𝓁 𝑘 (𝐺) ⟶ MH 𝓁−1 𝑘−1 (𝐺) between magnitude homologies of a digraph 𝐺, which make them chain complexes. Then we show that its homology  𝓁 𝑘 (𝐺) is non-trivial and homotopy invariant in the context of 'homotopy theory of digraphs' developed by Grigor'yan-Muranov-S.-T. Yau et al. (G-M-Ys in the following). It is remarkable that the diagonal part of our homology  𝑘 𝑘 (𝐺) is isomorphic to the reduced path homology H𝑘 (𝐺) also introduced by G-M-Ys. Further, we construct a spectral sequence whose first page is isomorphic to magnitude homology MH 𝓁 𝑘 (𝐺), and the second page is isomorphic to our homology  𝓁 𝑘 (𝐺). As an application, we show that the diagonality of magnitude homology implies triviality of reduced path homology. We also show that H𝑘 (g) = 0 for 𝑘 ⩾ 2 and H1 (g) ≠ 0 if any edges of an undirected graph g is contained in a cycle of length ⩾ 5.

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.

This paper studies the magnitude homology of graphs focusing mainly on the relationship between its diagonality and the girth. Magnitude and magnitude homology are formulations of the Euler characteristic and the corresponding homology, respetively, for finite metric spaces, first introduced by Leinster and Hepworth-Willerton. Several authors study them restricting to graphs with path metric, and some properties which are similar to the ordinary homology theory have come to light. However, the whole picture of their behavior is still unrevealed, and it is expected that they catch some geometric properties of graphs. In this article, we show that the girth of graphs partially determines magnitude homology, that is, the larger girth a graph has, the more homologies near the diagonal part vanish. Furthermore, applying this result to a typical random graph, we investigate how the diagonality of graphs varies statistically as the edge density increases. In particular, we show that there exists a phase transition phenomenon for the diagonality.

This paper studies the magnitude homology of graphs focusing mainly on the relationship between its diagonality and the girth. The magnitude and magnitude homology are formulations of the Euler characteristic and the corresponding homology, respectively, for finite metric spaces, first introduced by Leinster and Hepworth–Willerton. Several authors study them restricting to graphs with path metric, and some properties which are similar to the ordinary homology theory have come to light. However, the whole picture of their behaviour is still unrevealed, and it is expected that they catch some geometric properties of graphs. In this article, we show that the girth of graphs partially determines the magnitude homology, that is, the larger girth a graph has, the more homologies near the diagonal part vanish. Furthermore, applying this result to a typical random graph, we investigate how the diagonality of graphs varies statistically as the edge density increases. In particular, we show that there exists a phase transition phenomenon for the diagonality.

We determine the ring structure of the loop homology of some global quotient orbifolds. We can compute by our theorem the loop homology ring with suitable coefficients of the global quotient orbifolds of the form [M/G] for M being some kinds of homogeneous manifolds, and G being a finte subgroup of a path connected topological group G acting on M. It is shown that these homology rings split into the tensor product of the loop homology ring of the manifold H * (LM) and that of the classifying space of the finite group,which coincides with the center of the group ring Z(k[G]).