  Quantum states of spin systems that can be represented with weighted graphs
$G(V, E)$ are studied. The geometrical characteristics of these states are
examined. We find that the velocity of quantum evolution is determined by the
sum of the weighted degrees of the nodes in the graph, constructed by raising
to the second power the weights in $G(V, E)$. The curvature depends on the sum
of the weighted degrees of nodes in graphs constructed by raising the weights
in $G(V, E)$ to the second and fourth powers. It also depends on the sum of the
products of the weights of edges forming squares in graph $G(V, E)$. The
torsion in addition is related to the sum of the products of the weights of
edges in graph $G(V, E)$ forming triangles $S_3$. Geometric properties of
quantum graph states and the sum of the weighted degrees of nodes have been
calculated with quantum programming on IBM's quantum computer for the case of a
spin chain.
