From Laplacian-to-Adjacency matrix for continuous spins on graphs

Department of Physics, University of Trieste - Strada Costiera 11, 34151 Trieste, Italy

Received: 4 January 2026
Accepted: 6 March 2026

Abstract

The study of spins and particles on graphs has broad applications, from the dynamics of interacting systems on networks to combinatorial problems. Here, we study the large-n limit of the O(n) model on graphs, which is considerably more challenging than on regular lattices, as the loss of translational invariance gives rise to an infinite set of saddle point constraints in the thermodynamic limit. We show that the free energy at low and high temperature T is determined by the spectrum of two fundamental graph-theoretic objects: the Laplacian matrix at low T and the Adjacency matrix at high T. Their interplay is studied across several classes of graphs. For regular lattices the two coincide. We obtain an exact solution on trees, where the Lagrange multipliers interestingly depend solely on the number of nearest neighbors. We further contrast these classical results with those for a quantum spin model on an exemplary tree. For decorated lattices, the singular part of the free energy is governed by the Laplacian spectrum, whereas this is true for the full free energy only in the zero-temperature limit. Finally, we discuss a bipartite fully connected graph to highlight the importance of a finite coordination number in these results.