Loop Perturbed Regular Graphs

• Loop perturbed regular graphs are regular graphs modified by self-loops that alter spectral properties, dynamics, and combinatorial characteristics.
• These perturbations directly affect spectral density, gap closure, and random walk behavior, influencing mixing times and synchronization efficiency.
• The modifications also enable breakthroughs in Hamiltonicity and state transfer, offering a pathway to uniform spectral invariance in walk-regular environments.

Loop perturbed regular graphs comprise a class of structures arising from introducing self-loops or related local modifications to regular graphs, with ensuing effects on spectral properties, random walk dynamics, combinatorial expansion, and quantum transport. Recent research has clarified the precise role of walk-regularity, spectral invariants, and phase transitions associated with such perturbations, with connections to state transfer, synchronization, and Hamiltonicity in random and deterministic regimes.

1. Definitions, Constructions, and Structural Properties

A $d$-regular graph $G=(V,E)$ is perturbed by introducing loops of weight $\sigma \ge 0$ at a subset $\omega \subset V$ to yield a loop-perturbed regular graph, denoted $G + \sigma V_\omega$ with adjacency matrix

$$(A_\sigma)_{xy} = A_G(x,y) + \sigma \cdot \mathbf{1}_{x=y \in \omega}.$$

A key special case is the rank-one perturbation (loop at a single vertex $o \in V$), fundamental to phase transition analysis in random walks and spectral theory (Abert et al., 9 Dec 2025, Dalfó et al., 2012).

Alternatively, loop perturbations can be studied in ensembles such as $(\ell,k)$–Husimi graphs, where each vertex participates in exactly $k$ loops of length $\ell$, imposing a periodic or core structure and allowing explicit computation of spectral, dynamical, and combinatorial characteristics (Metz et al., 2011).

2. Spectral Theory and Walk-Regularity

A graph $G$ is walk-regular if for all $\ell \geq 0$, the number of closed walks of length $\ell$ rooted at any $u \in V$ is constant across $u$: $(A^\ell)_{uu} \ \text{independent of}\ u.$ This is equivalent to constancy of local multiplicities $m_u(\lambda_i)$ in the spectral decomposition, also called spectrum-regularity. For pairs of vertices at distance $h$, $h$-punctual walk-regularity requires the number of walks of length $\ell$ between $u,v$ at distance $h$ also depend only on $\ell$ and $h$ (Dalfó et al., 2012).

Key equivalence: For a walk-regular $G$, the characteristic polynomials of loop-addition, vertex-deletion, and pendant edge addition are vertex-independent and thus the resulting perturbed graphs are pairwise cospectral. This equivalence does not generally hold in the absence of walk-regularity, directly linking spectral invariance under loop perturbation to the uniformity of the graph’s walk structure.

For distance-regular graphs (maximal case of $h$-punctual walk-regularity), all such one- and two-vertex perturbations—loops, edge flips, vertex deletions—yield families of cospectral graphs for pairs or vertices at the same distance (Dalfó et al., 2012).

3. Spectral Density, Gap, and Dynamical Consequences

The spectral density $\rho(\lambda)$ of loop-perturbed regular graphs can be computed using generalized cavity or resolvent methods. For undirected $(\ell,k)$–Husimi graphs, the spectrum is described by

$$\rho(\lambda) = \lim_{\epsilon \to 0^+} \frac{1}{\pi} \Im \frac{1}{z - k G_s(z)}, \quad z = \lambda - i\epsilon,$$

where $G_s(z)$ is the root of a self-consistency equation that incorporates loops of length $\ell$ (Metz et al., 2011).

Impact of loops:

• The spectral gap $\Delta(\ell,k) = 2k - \lambda_{N-1}$ decreases monotonically as $\ell$ decreases; short loops (small $\ell$) close the gap, slow mixing (increase mixing times), and degrade expansion.
• The eigenratio $Q$ (relevant for synchronization) increases as more short loops are present, making stable synchronization more difficult.

For infinite $d$-regular trees, explicit formulas for both the spectral gap and $\sigma^*$, the critical loop weight for phase transitions in random walks, are available (Abert et al., 9 Dec 2025, Metz et al., 2011).

4. Phase Transitions, Random Walks, and Entropy Maximization

Upon addition of loop(s) at one or more vertices:

• For $\sigma < \sigma^*$, the Uniform Random Walk (URW) is transient and maximizes entropy per step at rate $\log d$.
• At the critical value $\sigma = \sigma^* = d / f_{oo}(1/d)$ (with $f_{oo}$ the Green function at the base vertex), recurrence transitions may occur, with existence criteria from the $\ell^2$-norm of the Green function.
• For $\sigma > \sigma^*$, the URW becomes localized near looped vertices, with positive recurrence and exponential decay of the stationary measure away from loops.

The operator-theoretic formalism for such phase transitions uses Sherman–Morrison–Woodbury type formulas for the perturbed resolvent, yields explicit transition probabilities, and allows for a complete classification of nature (delocalized vs. localized) of random walk behavior (Abert et al., 9 Dec 2025). In ergodic graphings with small enough loop weight, the principal eigenfunction and entropy maximization property are preserved.

5. Hamiltonicity, State Transfer, and Combinatorial Applications

Hamiltonicity via random loop perturbations: For any $d$-regular $G$ with $d \geq 3$, the addition of a random 2-factor (disjoint cycles covering all vertices) yields a Hamiltonian graph with high probability. The proof relies on Pósa rotations to extend short cycles and a tailored second-moment method to join long cycles, with critical use of expansion and randomness introduced by the 2-factor. The $d=2$ threshold remains open (Henderson et al., 26 Jun 2025).

Quantum and classical state transfer: In strongly regular graphs, loop perturbations (weighted loops and edges at $u,v$) parametrized by $(\beta,\gamma)$ facilitate both perfect and pretty good state transfer between $u,v$, provided spectral conditions such as strong cospectrality and parity constraints are met. Explicit spectral decompositions after perturbation distinguish subspaces supporting transfer and impose arithmetic constraints for transfer to occur. Infinite families of strongly regular graphs permit exact choices of parameters achieving perfect state transfer (Godsil et al., 2017).

6. Equivalences, Characterizations, and Extensions

The equivalence of distinct local perturbations (loop addition, vertex deletion, pendant attachment, edge manipulation) in their effect on the spectrum is sharply controlled by the degree of walk-regularity or spectrum-regularity of the graph. For $h$-punctually walk-regular graphs, cospectrality under such perturbations depends only on the distance $h$ between vertices or pairs, admitting uniform characterization of families of graphs perturbed at specific distances (Dalfó et al., 2012).

More generally, in the context of convergent sequences (Benjamini–Schramm limits), maximal entropy random walks (MERW) on finite loop-perturbed graphs converge to Uniform Random Walks on the limiting graphing, preserving spectral and entropy maximization structure in the presence of small loop perturbations (Abert et al., 9 Dec 2025).

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References

• "On perturbations of almost distance-regular graphs" (Dalfó et al., 2012)
• "Spectra of sparse regular graphs with loops" (Metz et al., 2011)
• "Hamilton cycles in regular graphs perturbed by a random 2-factor" (Henderson et al., 26 Jun 2025)
• "The Uniform Random Walk on graphs, loop processes and graphings" (Abert et al., 9 Dec 2025)
• "State transfer in strongly regular graphs with an edge perturbation" (Godsil et al., 2017)