  The article is devoted to the study of diagonal digraphs, i.e. digraphs whose
magnitude homology is concentrated on the diagonal. For a digraph $G$ and
$\ell\geq 2,$ we describe a condition, denoted by $(\mathcal{V}_\ell)$,
equivalent to vanishing of the second magnitude homology group ${\rm
MH}_{2,k}(G,\mathbb{Z})$ for all $k>\ell.$ In particular, diagonal digraphs
satisfy $(\mathcal{V}_2).$ As a corollary we obtain that the 2-dimensional
CW-complex obtained from a diagonal undirected graph by attaching 2-cells to
all squares and triangles of the graph is simply connected. We also give an
interpretation of diagonality in terms of Koszul algebras: a digraph $G$ is
diagonal if and only if the distance algebra $\sigma G$ is Koszul for any
ground field; and if and only if $G$ satisfies $(\mathcal{V}_2)$ and the path
cochain algebra $\Omega^\bullet(G)$ is Koszul for any ground field. To obtain a
source of examples of digraphs, we study the extended Hasse diagram $\hat G_K$
of a pure simplicial complex $K$. For a triangulation $K$ of a topological
manifold $M,$ we express the non-diagonal part of the magnitude homology of
$\hat G_K$ via the homology of $M$. As a corollary we obtain that, if $K$ is a
triangulation of a closed manifold $M$, then $\hat G_K$ is diagonal if and only
if $M$ is a homology sphere.
