Eigenvalues of the laplacian matrices of the cycles with one weighted edge

TL;DR

This work analyzes the spectrum of the Laplacian $L_{\alpha,n}$ of the cycle graph with a single weighted edge, proving that eigenvalues lie in $[0,4]$ and, for $0<\alpha<1$, are asymptotically distributed according to $g(x)=4\sin^2(x/2)$. The authors derive a main equation $x=d_{n,j}+\eta_\alpha(x)/n$ that governs nontrivial eigenvalues, establish their localization (odd $j$ yield exact $g((j-1)\pi/n)$ values while even $j$ lie between consecutive $g$-values), and develop fixed-point and Newton methods with rigorous convergence guarantees to compute them. They further obtain uniform asymptotic formulas for all eigenvalues with $error=O(1/n^3)$ and provide exact expressions and bounds for eigenvector norms. Comprehensive numerical experiments validate the theory and illustrate fast convergence of the proposed numerical schemes, with accurate eigenvalue approximations and stable eigenvector norms across parameter regimes. The work connects Toeplitz-perturbed spectral theory with practical computation, offering precise tools for diffusion and network dynamics on near-Toeplitz cycles.

Abstract

In this paper we study the eigenvalues of the laplacian matrices of the cyclic graphs with one edge of weight $α$ and the others of weight $1$. We denote by $n$ the order of the graph and suppose that $n$ tends to infinity. We notice that the characteristic polynomial and the eigenvalues depend only on $\operatorname{Re}(α)$. After that, through the rest of the paper we suppose that $0<α<1$. It is easy to see that the eigenvalues belong to $[0,4]$ and are asymptotically distributed as the function $g(x)=4\sin^2(x/2)$ on $[0,π]$. We obtain a series of results about the individual behavior of the eigenvalues. First, we describe more precisely their localization in subintervals of $[0,4]$. Second, we transform the characteristic equation to a form convenient to solve by numerical methods. In particular, we prove that Newton's method converges for every $n\ge3$. Third, we derive asymptotic formulas for all eigenvalues, where the errors are uniformly bounded with respect to the number of the eigenvalue.

Figures (5)

• Figure 1: Graph $G_{\alpha,7}$
• Figure 2: The left picture shows the left-hand side (green) and the right-hand side (blue) of \ref{['eq:main_eq']} for $\alpha=1/3$, $n=5$, $j=2,4$. The right picture corresponds to $\alpha=4/5$, $n=6$, $j=2,4,6$.
• Figure 3: Plots of $x\mapsto nx - (j-1)\pi$ (green) and $\eta_\alpha$ (blue), for $\alpha=1/3$, $n=5$ (left) and $\alpha=4/5$, $n=6$ (right).
• Figure 4: The left-hand side (green) and right-hand side (blue) of \ref{['eq:tangent_equality_L']} for $\alpha=0.7$ and $n=8$; the scales of the axes are different.
• Figure 5: The values $\vartheta_{\alpha,n,j}$ and $\lambda_{\alpha,n,j}$ for $\alpha=1/3$, $n=10$ (left) and $\alpha=4/5$, $n=6$ (right); the red marks on the horizontal axis correspond to $k\pi/10$, $1\le k\le 9$.

Theorems & Definitions (59)

• Theorem 1: eigenvalues' localization
• Theorem 2: main equation
• Theorem 3: convergence of Newton's method
• Theorem 4: asymptotic expansion of the eigenvalues
• Theorem 5: eigenvectors and their norms
• Proposition 6: the characteristic polynomial of $A_n$
• Corollary 7
• Proposition 8: eigenvectors of $A_n$
• proof
• Remark 9