[Paper Review] Some sharp bounds on the distance signless Laplacian spectral radius of graphs

Wenxi Hong, Lihua You|arXiv (Cornell University)|Aug 15, 2013

Graph theory and applications3 references3 citations

TL;DR

This paper establishes sharp upper and lower bounds for the distance signless Laplacian spectral radius of connected graphs using spectral graph theory. By deriving general bounds for nonnegative irreducible matrices and applying them to the distance signless Laplacian matrix, the authors provide tight spectral radius estimates based on vertex transmissions and second distance degrees, correcting a previously overlooked defect in an established bound for the standard signless Laplacian spectral radius.

ABSTRACT

M. Aouchiche and P. Hansen proposed the distance Laplacian and the distance signless Laplacian of a connected graph [Two Laplacians for the distance matrix of a graph, LAA 439 (2013) 21{33]. In this paper, we obtain three theorems on the sharp upper bounds of the spectral radius of a nonnegative matrix, then apply these theorems to signless Laplacian matrices and the distance signless Laplacian matrices to obtain some sharp bounds on the spectral radius, respectively. We also proposed a known result about the sharp bound of the signless Laplacian spectral radius has a defect.

Motivation & Objective

• To establish sharp upper and lower bounds for the spectral radius of the distance signless Laplacian matrix of connected graphs.
• To correct a previously published result on the signless Laplacian spectral radius that contains a defect in its sharpness condition.
• To extend spectral bounds from standard signless Laplacian matrices to distance-based analogues using transmission and second distance degree sequences.
• To provide computable, tight bounds for the distance signless Laplacian spectral radius using graph invariants such as vertex transmissions and sum of distances.

Proposed method

• Derive three general theorems on the spectral radius of nonnegative irreducible matrices, focusing on row sum and matrix perturbation properties.
• Apply these theorems to the signless Laplacian matrix to recover and correct known bounds, identifying a flaw in a previously cited result.
• Introduce the distance signless Laplacian matrix $\mathcal{D}^Q = \mathrm{Tr}(G) + \mathcal{D}(G)$, where $\mathrm{Tr}(G)$ is the diagonal matrix of vertex transmissions.
• Use the Perron–Frobenius theorem to ensure the largest eigenvalue $\delta_1^Q(G)$ is simple and positive with a positive eigenvector.
• Establish bounds via similarity transformations: $\mathrm{Tr}(G)^{-1}\mathcal{D}^Q\mathrm{Tr}(G)$, whose row sums yield $\mathcal{D}_i + \frac{T_i}{\mathcal{D}_i}$.
• Derive bounds based on the squared matrix $ (\mathcal{D}^Q)^2 $, whose row sums are $ 2T_i + 2\mathcal{D}_i^2 $, leading to bounds involving $ \sqrt{2T_i + 2\mathcal{D}_i^2} $.

Experimental results

Research questions

• RQ1 What are the tightest possible upper and lower bounds for the distance signless Laplacian spectral radius in terms of graph invariants?
• RQ2 How can general bounds on the spectral radius of nonnegative matrices be specialized to yield sharp results for the distance signless Laplacian?
• RQ3 What is the flaw in the previously claimed sharp bound for the standard signless Laplacian spectral radius, and how can it be corrected?
• RQ4 Under what graph conditions do the derived bounds for $\delta_1^Q(G)$ become equality?
• RQ5 Can the spectral radius of $\mathcal{D}^Q$ be bounded using only vertex transmission and second distance degree sequences?

Key findings

• The paper establishes that $ \delta_1^Q(G) \leq \max\left\{ \mathcal{D}_i + \frac{T_i}{\mathcal{D}_i} \right\} $ and $ \delta_1^Q(G) \geq \min\left\{ \mathcal{D}_i + \frac{T_i}{\mathcal{D}_i} \right\} $, with equality if and only if all values $ \mathcal{D}_i + \frac{T_i}{\mathcal{D}_i} $ are equal.
• A previously cited sharp bound for the signless Laplacian spectral radius is shown to be incorrect due to an invalid equality condition, and a corrected version is provided.
• The bound $ \delta_1^Q(G) \leq \max\left\{ \sqrt{2T_i + 2\mathcal{D}_i^2} \right\} $ is established, with equality if and only if $ T_i + \mathcal{D}_i^2 $ is constant across all vertices or under specific reducibility conditions on $ (\mathcal{D}^Q)^2 $.
• For the complete graph $ K_n $, the distance signless Laplacian spectral radius is exactly $ 2n - 2 $, and this value serves as a known lower bound.
• The equality conditions in the derived bounds are fully characterized: for the $ \mathcal{D}_i + \frac{T_i}{\mathcal{D}_i} $ bound, equality holds iff all such values are equal; for the $ \sqrt{2T_i + 2\mathcal{D}_i^2} $ bound, equality holds under specific structural conditions on the graph or matrix reducibility.
• The authors demonstrate that the spectral radius of $ \mathcal{D}^Q $ is tightly controlled by local graph invariants—vertex transmissions and second distance degrees—enabling efficient estimation without full eigenvalue computation.