[Paper Review] On Distribution of Laplacian Eigenvalues of Graphs

Bilal Ahmad Rather|arXiv (Cornell University)|Jul 19, 2021

Graph theory and applications62 references4 citations

TL;DR

This doctoral thesis investigates the distribution of Laplacian eigenvalues across various graph matrices, including the Laplacian, normalized Laplacian, signless Laplacian, and distance signless Laplacian. It establishes upper bounds for the sum of the k largest Laplacian eigenvalues, proves Brouwer’s conjecture for broader graph classes, verifies the Laplacian energy conjecture for trees of diameter 4 and those with few non-pendent vertices, and derives exact spectra and energy bounds for power graphs and generalized distance matrices, particularly showing the star graph minimizes generalized distance energy under specific α conditions.

ABSTRACT

The work in this thesis concerns the investigation of eigenvalues of the Laplacian matrix, normalized Laplacian matrix, signless Laplacian matrix and distance signless Laplacian matrix of graphs. In Chapter 1, we present a brief introduction of spectral graph theory with some definitions. Chapter $2$ deals with the sum of $ k $ largest Laplacian eigenvalues $ S_{k}(G) $ of graph $ G $ and Brouwer's conjecture. We obtain the upper bounds for $ S_{k}(G) $ for some classes of graphs and use them to verify Brouwer's conjecture for these classes of graphs. Also, we prove Brouwer's conjecture for more general classes of graphs. In Chapter $3$, we investigate the Laplacian eigenvalues of graphs and the Laplacian energy conjecture for trees. We prove the Laplacian energy conjecture completely for trees of diameter $ 4 $. Further, we prove this conjecture for all trees having at most $ \frac{9n}{25}-2 $ non-pendent vertices. Also, we obtain the sufficient conditions for the truth of conjecture for trees of order $ n $. In Chapter $4$, we determine the normalized Laplacian spectrum of the joined union of regular graphs and obtain the spectrum of some well known graphs. As consequences of joined union, we obtain the normalized Laplacian spectrum of power graphs associated to finite cyclic groups. In Chapter $5$, we find the distance signless Laplacian spectrum of regular graphs and zero-divisor graphs associated to finite commutative ring. Also, we find the bounds for spectral radius of generalized distance matrix. Further, we obtain the generalized distance energy for bipartite graphs and trees. We prove that the complete bipartite graph has minimum generalized distance energy among all connected bipartite graphs. Besides, for $ α\in \big(0, \frac{2n}{3n-2}\big) $, we show that the star graph has minimum generalized distance energy among all trees.

Motivation & Objective

• To investigate the distribution of Laplacian eigenvalues across various graph matrices, including Laplacian, normalized Laplacian, signless Laplacian, and distance signless Laplacian matrices.
• To verify Brouwer’s conjecture on the sum of the k largest Laplacian eigenvalues for broader classes of graphs.
• To prove the Laplacian energy conjecture for trees of diameter 4 and for trees with at most (9n/25)−2 non-pendent vertices.
• To determine the normalized Laplacian spectrum of joined unions of regular graphs and apply it to power graphs of finite cyclic groups.
• To derive bounds for the spectral radius and generalized distance energy of bipartite graphs and trees, and identify extremal graphs minimizing energy.

Proposed method

• Derives upper bounds for the sum of the k largest Laplacian eigenvalues, S_k(G), using structural properties of graphs.
• Applies spectral bounds and eigenvalue interlacing to verify Brouwer’s conjecture for specific graph families.
• Uses the generalized distance matrix D_α(G) = αD(G) + (1−α)A(G), where D(G) is the distance matrix and A(G) the adjacency matrix.
• Employs the generalized distance energy formula E^D_α(T) = 2 max_{1≤j≤n} {∑_{i=1}^j ∂_i(T) − (2αjW(T))/n}, with W(T) being the Wiener index.
• Applies extremal graph theory and eigenvalue majorization to identify graphs minimizing generalized distance energy.
• Uses spectral decomposition and matrix analysis to compute normalized Laplacian spectra of joined unions and power graphs.

Experimental results

Research questions

• RQ1 For which classes of graphs does Brouwer’s conjecture on the sum of the k largest Laplacian eigenvalues hold, and can it be extended to more general families?
• RQ2 Does the Laplacian energy conjecture hold for all trees of diameter 4, and what structural conditions ensure its validity for trees with few non-pendent vertices?
• RQ3 What is the exact normalized Laplacian spectrum of the joined union of regular graphs, and how does it apply to power graphs of finite cyclic groups?
• RQ4 What are the bounds for the spectral radius of the generalized distance matrix D_α(G), and which graphs minimize generalized distance energy?
• RQ5 For which values of α is the star graph S_n the minimizer of generalized distance energy among all trees of order n?

Key findings

• The Laplacian energy conjecture is fully proven for all trees of diameter 4.
• The conjecture holds for all trees with at most (9n/25)−2 non-pendent vertices.
• For α ∈ (0, 2n/(3n−2)), the star graph S_n minimizes generalized distance energy among all trees of order n.
• For α ∈ [2n/(3n−2), 1), the generalized distance energy of S_n is given by a closed-form expression involving a square root term dependent on n and α.
• The normalized Laplacian spectrum of the joined union of regular graphs is derived, and applied to determine the spectrum of power graphs of finite cyclic groups.
• The generalized distance energy of the complete bipartite graph is shown to be minimal among all connected bipartite graphs.