  We consider the skeleton of the pyramidal tours polytope. Hamiltonian tour is
called pyramidal if the salesperson starts in city $1$, then visits some cities
in increasing order, reaches city $n$ and returns to city $1$, visiting the
remaining cities in decreasing order. The polytope $PYR(n)$ is defined as the
convex hull of characteristic vectors of all pyramidal tours in the complete
graph $K_{n}$. The skeleton of the polytope $PYR(n)$ is the graph whose vertex
set is the vertex set of $PYR(n)$ and edge set is the set of geometric edges or
one-dimensional faces of $PYR(n)$. We describe the necessary and sufficient
condition for the adjacency of vertices of the polytope $PYR(n)$. On this basis
we developed an algorithm to check the vertex adjacency with a linear
complexity. We establish that the diameter of $PYR(n)$ skeleton equals 2, and
the asymptotically exact estimate of $PYR(n)$ skeleton's clique number is
$\Theta(n^{2})$. It is known that this value characterizes the time complexity
in a broad class of algorithms based on linear comparisons.
