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Monotone grid classes of permutations have proven very effective in helpingto determine structural and enumerative properties of classical permutationpattern classes. Associated with grid class $\mathrm{Grid}(M)$ is a graph,$G(M)$, known as its "row-column" graph. We prove that the exponential growthrate of $\mathrm{Grid}(M)$ is equal to the square of the spectral radius of$G(M)$. Consequently, we utilize spectral graph theoretic results tocharacterise all slowly growing grid classes and to show that for every$\gamma\geqslant2+\sqrt{5}$ there is a grid class with growth rate arbitrarilyclose to $\gamma$. To prove our main result, we establish bounds on the size ofcertain families of tours on graphs. In the process, we prove that the familyof tours of even length on a connected graph grows at the same rate as thefamily of "balanced" tours on the graph (in which the number of times an edgeis traversed in one direction is the same as the number of times it istraversed in the other direction).

Growth rates of permutation grid classes, tours on graphs, and the spectral radius