  Let $V$ be a finite dimensional vector space equipped with a non-degenerate
Hermitian form over a field $\mathbb{K}$. Let $\mathcal{G}(V)$ be the graph
with vertex set the $1$-dimensional non-degenerate subspaces of $V$ and
adjacency relation given by orthogonality. We give a complete description of
when $\mathcal{G}(V)$ is connected in terms of the dimension of $V$ and the
size of the ground field $\mathbb{K}$. Furthermore, we prove that if $\dim(V) >
4$ then the clique complex $\mathcal{F}(V)$ of $\mathcal{G}(V)$ is simply
connected. For finite fields $\mathbb{K}$, we also compute the eigenvalues of
the adjacency matrix of $\mathcal{G}(V)$. Then by Garland's method, we conclude
that $\tilde{H}_m(\mathcal{F}(V);\mathbb{k}) = 0$ for all $0\leq m\leq
\dim(V)-3$, where $\mathbb{k}$ is a field of characteristic $0$, provided that
$\dim(V)^2 \leq |\mathbb{K}|$. Under these assumptions, we deduce that the
barycentric subdivision of $\mathcal{F}(V)$ deformation retracts to the order
complex of the certain rank selection of $\mathcal{F}(V)$ which is
Cohen-Macaulay over $\mathbb{k}$.
  Finally, we apply our results to the Quillen poset of elementary abelian
$p$-subgroups of a finite group and to the study of geometric properties of the
poset of non-degenerate subspaces of $V$ and the poset of orthogonal
decompositions of $V$.
