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Some Laplacian eigenvalues can be computed by matrix perturbation

Piet Van Mieghem, Yingyue Ke

TL;DR

This work develops analytic perturbation-based methods to estimate Laplacian eigenvalues of graphs near a node with a unique degree $d_q$. It derives a perturbation Taylor series with explicit coefficients and introduces an Euler $t$-transformed series to extend convergence beyond the Taylor radius; numerical experiments show the Euler transform can converge where the Taylor series does not, depending on graph structure and the tuning parameter $t$. The authors provide explicit closed-form coefficients for the almost-regular case (one high-degree node) and demonstrate convergence behavior on trees, Erdős–Rényi graphs, and antiregular graphs, highlighting how degree differences and $t$ influence accuracy. The results offer a practical analytic tool that complements existing spectral bounds and sheds light on when perturbation-based eigenvalue estimates are reliable in spectral graph theory.

Abstract

Based on matrix perturbation theory, closed-form analytic expansions are studied for a Laplacian eigenvalue of an undirected, possibly weighted graph, which is close to a unique degree in that graph. An approximation is presented to provide an analytic estimate of a Laplacian eigenvalue and complements bounds on Laplacian eigenvalues in spectral graph theory. Then, we apply the Euler summation and numerically analyze how the structure of graphs influences the convergence of the corresponding Euler series. Moreover, we obtain the explicit form of the perturbation Taylor series and its Euler series of almost regular graphs in which only one node has a unique degree and all remaining nodes have the same degree. We find that the Euler series possesses a superior convergence range than the perturbation Taylor series for almost regular graphs.

Some Laplacian eigenvalues can be computed by matrix perturbation

TL;DR

This work develops analytic perturbation-based methods to estimate Laplacian eigenvalues of graphs near a node with a unique degree . It derives a perturbation Taylor series with explicit coefficients and introduces an Euler -transformed series to extend convergence beyond the Taylor radius; numerical experiments show the Euler transform can converge where the Taylor series does not, depending on graph structure and the tuning parameter . The authors provide explicit closed-form coefficients for the almost-regular case (one high-degree node) and demonstrate convergence behavior on trees, Erdős–Rényi graphs, and antiregular graphs, highlighting how degree differences and influence accuracy. The results offer a practical analytic tool that complements existing spectral bounds and sheds light on when perturbation-based eigenvalue estimates are reliable in spectral graph theory.

Abstract

Based on matrix perturbation theory, closed-form analytic expansions are studied for a Laplacian eigenvalue of an undirected, possibly weighted graph, which is close to a unique degree in that graph. An approximation is presented to provide an analytic estimate of a Laplacian eigenvalue and complements bounds on Laplacian eigenvalues in spectral graph theory. Then, we apply the Euler summation and numerically analyze how the structure of graphs influences the convergence of the corresponding Euler series. Moreover, we obtain the explicit form of the perturbation Taylor series and its Euler series of almost regular graphs in which only one node has a unique degree and all remaining nodes have the same degree. We find that the Euler series possesses a superior convergence range than the perturbation Taylor series for almost regular graphs.
Paper Structure (21 sections, 125 equations, 9 figures)

This paper contains 21 sections, 125 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Theory
  3. Brief review of matrix perturbation theory
  4. Coefficients and the perturbation Taylor series
  5. Convergence of the perturbation Taylor series
  6. Euler summation and the Euler series
  7. Extension of the Euler series and analogy with Lagrange series
  8. Performance evaluation of the Euler series $\xi_{q;K}$
  9. Impact of the tuning parameter $t$ on the convergence of the Euler series $\xi_{q;K}$
  10. Impact of the node degree on the convergence of the Euler series $\xi_{q;K}$
  11. Impact of the degree difference $\kappa$ on the convergence of $\xi_{q;K}$
  12. Almost regular graph with one high degree node
  13. Deduction of the general form for the coefficient $c_{j}$
  14. Closed form for the coefficient $c_{m}$
  15. Perturbed eigenvalue $\xi_{q}\left( \zeta\right)$ of the matrix $\Delta+\zeta A$
  16. ...and 6 more sections

Figures (9)

  • Figure 1: The accuracy $\alpha$ of the Euler $t$-series $\xi_{q; K}$ versus the number of terms $K$ and tuning parameter $t$ for two nodes with a unique degree in Example 1.
  • Figure 2: The accuracy $\alpha$ of the Euler $t$-series $\xi_{q; K}$ versus the number of terms $K$ and tuning parameter $t$ for four nodes with a unique degree in Example 2.
  • Figure 3: The accuracy $\alpha$ versus the number of terms $K$ and tuning parameter $t$ for each node with a unique degree in Example 3.
  • Figure 4: An example of a graph $G_{(N,k)}$ with $N=8$ nodes and $k=1$.
  • Figure 5: The accuracy $\alpha$ of the Euler $t$-series $\xi_{31;K}$ of the node $q=31$ with the largest degree $d_{31}=30$ versus the number of terms $K$ and tuning parameter $t$ for each parameter $k$ in graphs $G_{(31, k)}$.
  • ...and 4 more figures