We present a Markov chain Monte Carlo (MCMC) method to make a geometric graph that 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 dij - α , where dij is the distance between the two vertices and α 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 α.