On three conjectures involving the signless Laplacian spectral radius of graphs

The Q-index of a simple graph is the largest eigenvalue of its signless Laplacian, or Qmatrix. In our previous paper [1] we gave three lower and three upper bounds for the Q-index of nested split graphs, also known as threshold graphs. In this paper, we give another two upper bounds, which are expressed as solutions of cubic equations (in contrast to quadratics from [1]). Some computational results are also included.

show abstract

“…Applying Lemma 3.1(1) and the inequality κ ≤ ν − 1 +d (conjectured in [4] and first proved in [9] and independently in [2]) we first obtain…”

Section: Proposition 32mentioning

confidence: 79%

Some further bounds for the Q-index of nested split graphs

et al. 2012

View full text Add to dashboard Cite

show abstract

“…Recently there is a lot of work on the spectral radius or the signless Laplacian spectral radius of graphs, see [6,13,14,15,17,19,24,26,28,32,36,40,41,45,49] et al Some investigation on graphs with prefect matching or with given matching number is an important topic in the theory of graph spectra, see [7, 8, [32] determined the unique graph with the maximum signless Laplacian spectral radius among all cacti in ℓ m n with n = 2m, and gave a conjecture about the case n ≥ 2m + 1 as follows.…”

Section: Introductionmentioning

confidence: 99%

On a conjecture for the signless Laplacian spectral radius of cacti with given matching number

Shen

You

Zhang

et al. 2016

Linear and Multilinear Algebra

View full text Add to dashboard Cite

A connected graph G is a cactus if any two of its cycles have at most one common vertex. Let ℓ m n be the set of cacti on n vertices with matching number m. S.C. Li and M.J. Zhang determined the unique graph with the maximum signless Laplacian spectral radius among all cacti in ℓ m n with n = 2m. In this paper, we characterize the case n ≥ 2m+1. This confirms the conjecture of Li and Zhang (S.C. Li, M.J. Zhang, On the signless Laplacian index of cacti with a given number of pendant vetices, Linear Algebra Appl. 436, 2012,[4400][4401][4402][4403][4404][4405][4406][4407][4408][4409][4410][4411]. Further, we characterize the unique graph with the maximum signless Laplacian spectral radius among all cacti on n vertices.

show abstract

“…The arguments given in [8,10,35] support the above idea. The second reason for the interest in the signless Laplacian matrix is that using Q-spectra to study graphs is more efficient than studying them by their (adjacency) spectra; see S. Li (B) · L. Zhang Faculty of Mathematics and Statistics, Central China Normal University, Wuhan 430079, People's Republic of China e-mail: lscmath@mail.ccnu.edu.cn [5,8,9,[11][12][13][14][16][17][18]20,21,30,37,38]. Another reason for the interest in the signless Laplacian matrix is that the set of signless Laplacian matrices is a class of non-negative matrices each of which has combinatorial significance.…”

Section: Introductionmentioning

confidence: 99%