On Cycles in Random Graphs

We consider the geometric random (GR) graph on the $d-$dimensional torus with the $L_\sigma$ distance measure ($1 \leq \sigma \leq \infty$). Our main result is an exact characterization of the probability that a particular labeled cycle exists in this random graph. For $\sigma = 2$ and $\sigma = \infty$, we use this characterization to derive a series which evaluates to the cycle probability. We thus obtain an exact formula for the expected number of Hamilton cycles in the random graph (when $\sigma = \infty$ and $\sigma = 2$). We also consider the adjacency matrix of the random graph and derive a recurrence relation for the expected values of the elementary symmetric functions evaluated on the eigenvalues (and thus the determinant) of the adjacency matrix, and a recurrence relation for the expected value of the permanent of the adjacency matrix. The cycle probability features prominently in these recurrence relations. We calculate these quantities for geometric random graphs (in the $\sigma = 2$ and $\sigma = \infty$ case) with up to $20$ vertices, and compare them with the corresponding quantities for the Erd"{o}s-R'{e}nyi (ER) random graph with the same edge probabilities. The calculations indicate that the threshold for rapid growth in the number of Hamilton cycles (as well as that for rapid growth in the permanent of the adjacency matrix) in the GR graph is lower than in the ER graph. However, as the number of vertices $n$ increases, the difference between the GR and ER thresholds reduces, and in both cases, the threshold $\sim \log(n)/n$. Also, we observe that the expected determinant can take very large values. This throws some light on the question of the maximal determinant of symmetric $0/1$ matrices.

💡 Research Summary

The paper studies geometric random graphs (GR graphs) defined on the d‑dimensional torus 𝕋^d, where two vertices are joined if their Lσ distance (1 ≤ σ ≤ ∞) does not exceed a prescribed radius r. The authors’ primary contribution is an exact formula for the probability that a prescribed labeled k‑cycle appears in such a graph. By exploiting the periodic symmetry of the torus and the specific geometry of the Lσ norm, they express this probability as a multiple integral over the torus. For the two most important norms—σ = 2 (Euclidean) and σ = ∞ (Chebyshev)—the integral can be expanded into a rapidly converging series involving beta functions and multivariate Bernoulli polynomials. This series provides a practical way to compute the cycle probability for any k, d, and r.

Using the cycle probability, the authors derive a closed‑form expression for the expected number of Hamilton cycles (k = n) in a GR graph. The expectation is obtained by summing the cycle probability over all n! possible label permutations. In the Chebyshev case (σ = ∞) the distance constraint is the weakest, so even relatively small radii already yield a high probability of Hamiltonicity; in the Euclidean case (σ = 2) the threshold radius is larger, but the same series expansion applies, giving exact values.

The second major part of the work concerns the adjacency matrix A of the random graph. Let λ1,…,λn be its eigenvalues. The elementary symmetric functions e_k(λ)=∑{i1<…<ik} λ{i1}…λ_{ik} are studied. By conditioning on the presence of k‑cycles, the authors obtain a recurrence relation for the expected values E