We study a minimal model of disordered systems, the random field Ising model (RFIM) on a generalized Petersen Graph, $GP(N,k)$, which consists of an interconnected outer and inner loop with $N$ nodes in each loop. Each node in $GP(N,k)$ has a coordination number of $z=3$, and by varying the parameter $k$ different connections between the nodes in the inner loop can be obtained. The objective is to study whether different connectivities between nodes in these graphs affect the system's response to an external field when the coordination number is fixed. This is of interest because critical behavior is absent for $z\le 3$ on a random graph which has been solved exactly as well as on the honeycomb lattice in the context of RFIM. Using single-spin-flip Glauber dynamics at zero temperature, we compare the system's response with the known case of a $z=3$ random graph and the generalized Petersen graph for various connectivities $k$, albeit for the same $z$. Our study finds the absence of critical behavior on $GP(N,k)$, highlighting the importance of the coordination number over varying connectivities between the nodes. Additionally, we explore the case of directed $GP(N,k)$ and compare it with the undirected $GP(N,k)$ results.

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• Nonequilibrium statistical mechanics
• Phase transitions