[Paper Review] Upper bound for the Laplacian eigenvalues of a graph

Miriam Farber, Ido Kaminer|arXiv (Cornell University)|Jun 3, 2011

Graph theory and applications5 references3 citations

TL;DR

This paper establishes a new upper bound for the Laplacian eigenvalues of unweighted and weighted graphs, showing that $\lambda_{m+1}(G) \leq d_m(G) + m - 1$ for unweighted graphs when the complement $\bar{G}$ is not $K_m + (n-m)K_1$. The bound is derived using spectral duality and degree sequence analysis, and extended to weighted graphs via normalization and Gersgorin-type eigenvalue estimates, offering tighter interval constraints on eigenvalues than prior results in certain cases.

ABSTRACT

In this note we give a new upper bound for the Laplacian eigenvalues of an unweighted graph. Let $G$ be a simple graph on $n$ vertices. Let $d_{m}(G)$ and $λ_{m+1}(G)$ be the $m$-th smallest degree of $G$ and the $m+1$-th smallest Laplacian eigenvalue of $G$ respectively. Then $ λ_{m+1}(G)\leq d_{m}(G)+m-1 $ for $\bar{G} eq K_{m}+(n-m)K_1 $. We also introduce upper and lower bound for the Laplacian eigenvalues of weighted graphs, and compare it with the special case of unweighted graphs.

Motivation & Objective

• To derive a new upper bound for the $(m+1)$-th smallest Laplacian eigenvalue of an unweighted graph based on its degree sequence.
• To extend the upper bound to weighted graphs by normalizing edge weights and analyzing the complement graph's structure.
• To compare the new bounds with existing lower and upper bounds, particularly those from Brouwer-Haemers and Grone-Merris.
• To characterize the conditions under which equality holds in the derived bounds, especially in star-like graphs with added edges.

Proposed method

• Use spectral duality via the complement graph $\bar{G}$ and the identity $\lambda_i(\bar{G}) = n - \lambda_{n-i+2}(G)$ to relate eigenvalues of $G$ and $\bar{G}$.
• Apply Brouwer-Haemers' lower bound on $\lambda_{n-m+1}(\bar{G})$ to derive a lower bound on $\lambda_{m+1}(G)$ via duality.
• Establish the key inequality $\lambda_{m+1}(G) \leq d_m(G) + m - 1$ under the condition $\bar{G} \neq K_m + (n-m)K_1$.
• For weighted graphs, normalize edge weights by the maximum weight $a$, then apply the same duality and eigenvalue estimation techniques to the normalized graph $\hat{G}$.
• Use the Gersgorin Disc Theorem to bound the quadratic form $\langle A(G_{l_m})g, g \rangle / \langle g, g \rangle \leq \Delta(G_{l_m})$ in the Rayleigh quotient for lower eigenvalue bounds.
• Derive the upper bound $\lambda_{m+1}(G) \leq d_m(G) + m a - \delta(G_{S_m})$ for weighted graphs by analyzing the induced subgraph $H$ on the $m$ largest-degree vertices in $\bar{\hat{G}}$.

Experimental results

Research questions

• RQ1 What is the tightest possible upper bound for the $(m+1)$-th Laplacian eigenvalue of an unweighted graph in terms of its degree sequence?
• RQ2 How do the Laplacian eigenvalues of weighted graphs relate to the maximum edge weight and the structure of high-degree vertex subgraphs?
• RQ3 Under what conditions does the derived upper bound become tight, and how does this compare to known lower bounds?
• RQ4 Can spectral duality between a graph and its complement be used to derive new eigenvalue inequalities for Laplacian matrices?

Key findings

• The upper bound $\lambda_{m+1}(G) \leq d_m(G) + m - 1$ holds for unweighted graphs as long as $\bar{G} \neq K_m + (n-m)K_1$, providing a significant refinement over previous bounds.
• For weighted graphs, the upper bound is $\lambda_{m+1}(G) \leq d_m(G) + m a - \delta(G_{S_m})$, where $a$ is the maximum edge weight and $\delta(G_{S_m})$ is the minimum degree in the subgraph induced by the $m$ vertices of highest degree.
• Equality in the unweighted bound occurs when $d_m(G) = d_{m-1}(G) + n - 3$, indicating a specific structural condition on the degree sequence.
• In the unweighted case, the derived bounds $d_m(G) - n + m + 1 \leq \lambda_m(G) \leq d_{m-1}(G) + m - 2$ are tighter than the general weighted case bounds, which are $d_m(G) - n + m \leq \lambda_m(G) \leq d_{m-1}(G) + m - 1$.
• The bound is sharp for graphs constructed by adding edges to a star $K_{1,n-1}$ such that $d_{n-1}(G) = 2$, where equality holds for $\lambda_n(G)$.
• The method successfully generalizes known eigenvalue bounds using duality and subgraph degree analysis, offering a systematic framework for bounding Laplacian eigenvalues.