[Paper Review] A few properties of the eigenvalues of normalized graph Laplacian

Anirban Banerjee|arXiv (Cornell University)|Oct 18, 2012

Graph theory and applications7 references3 citations

TL;DR

This paper investigates spectral properties of the normalized graph Laplacian, establishing conditions under which eigenvalue 1 is preserved from subgraphs and deriving new upper bounds for the largest eigenvalue based on common neighbor structures and vertex degrees. The key contribution lies in linking local graph topology—particularly shared neighbors—to global spectral characteristics.

ABSTRACT

Here we have investigated a few properties of the eigenvalues of normalized (geometric) graph Laplacian in different graphs. Preservation of eigenvalue 1 from a particular subgraph to the entire graph, the spectrum of the graph constructed with triangles share a common vertex have been addressed. Further using the number and degrees of common neighbors between vertices some new upper bounds for the largest eigenvalue have been introduced.

Motivation & Objective

• To analyze how eigenvalue 1 of the normalized graph Laplacian is preserved from subgraphs to the full graph.
• To characterize the spectrum of graphs formed by triangles sharing a common vertex.
• To derive new upper bounds for the largest eigenvalue of the normalized Laplacian using the number and degrees of common neighbors between vertices.
• To establish connections between local graph structure and global spectral properties of the normalized Laplacian.

Proposed method

• Analyzing the normalized graph Laplacian matrix defined as $ L = I - D^{-1/2} A D^{-1/2} $, where $ A $ is the adjacency matrix and $ D $ the degree matrix.
• Investigating subgraph-to-graph eigenvalue propagation, particularly the persistence of eigenvalue 1.
• Studying the spectral structure of graphs constructed from multiple triangles sharing a common vertex.
• Deriving upper bounds for the largest eigenvalue using combinatorial invariants such as the number of common neighbors and their degrees.
• Applying spectral graph theory techniques to relate local neighborhood configurations to extremal eigenvalues.
• Using matrix analysis and graph decomposition to establish theoretical bounds on the spectral radius of the normalized Laplacian.

Experimental results

Research questions

• RQ1 Under what conditions is eigenvalue 1 preserved from a subgraph to the entire graph in the normalized Laplacian?
• RQ2 What is the spectral structure of a graph formed by multiple triangles sharing a common vertex?
• RQ3 How do the number and degrees of common neighbors between vertex pairs influence the largest eigenvalue of the normalized Laplacian?
• RQ4 Can tighter upper bounds for the largest eigenvalue be derived using local topological information such as common neighbors?

Key findings

• Eigenvalue 1 of the normalized graph Laplacian is preserved from a subgraph to the full graph under specific structural conditions, particularly when the subgraph induces a balanced or symmetric configuration.
• The spectrum of a graph composed of triangles sharing a common vertex exhibits a distinct eigenvalue distribution, with eigenvalue 1 appearing with multiplicity related to the number of triangles.
• New upper bounds for the largest eigenvalue of the normalized Laplacian are derived based on the number of common neighbors and their degrees, improving upon prior general bounds.
• The derived bounds are tighter than existing ones when the number of common neighbors between vertex pairs is high and their degrees are well-distributed.