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

An Efficient Monte-Carlo Method to Make a Geometric Graph with a Fixed Connectivity