  We present a Markov chain Monte-Carlo (MCMC) method to make a geometric graph
which satisfies the following two conditions: (i) The degree of each vertex is
fixed to a positive integer $k$. (ii) The probability that two vertices located
on a $d$-dimensional hypercubic lattice are connected by an edge is
proportional to $d_{ij}^{-\alpha}$, where $d_{ij}$ is the distance between the
two vertices and $\alpha$ is a positive exponent. We introduce a reverse update
method and a list-based update method for the MCMC method. The graph is updated
efficiently by the MCMC method since the two update methods work
complementarily. We also investigate a ferromagnetic Ising model defined on the
geometric graph as a test case. As a result, we have confirmed that the nature
of ferromagnetic transition significantly depends on the exponent $\alpha$.
