  Geometric properties of evolutionary graph states of spin systems generated
by the operator of evolution with Ising Hamiltonian are examined, using their
relationship with fluctuations of energy. We find that the geometric
characteristics of the graph states depend on properties of the corresponding
graphs. Namely, it is obtained that the fluctuations of energy in graph states
and therefore the velocity of quantum evolution, the curvature and the torsion
of the states are related with the total number of edges, triangles and squares
in the corresponding graphs. The obtained results give a possibility to
quantify the number of edges, triangles and squares in a graph on a quantum
devise and achieve quantum supremacy in solving this problem with the
development of a multi-qubit quantum computer. Geometric characteristics of
graph states corresponding to a chain, a triangle, and a square are detected on
the basis of calculations on IBM's quantum computer ibmq-manila.
