Taiwanese Journal of Mathematics

Open Problem on $\sigma$-invariant

Kinkar Ch. Das and Seyed Ahmad Mojallal

Full-text: Open access

Abstract

Let $G$ be a graph of order $n$ with $m$ edges. Also let $\mu_1 \geq \mu_2 \geq \cdots \geq \mu_{n-1} \geq \mu_n = 0$ be the Laplacian eigenvalues of graph $G$ and let $\sigma = \sigma(G)$ ($1 \leq \sigma \leq n$) be the largest positive integer such that $\mu_{\sigma} \geq 2m/n$. In this paper, we prove that $\mu_2(G) \geq 2m/n$ for almost all graphs. Moreover, we characterize the extremal graphs for any graphs. Finally, we provide the answer to Problem 3 in [8], that is, the characterization of all graphs with $\sigma = 1$.

Article information

Source
Taiwanese J. Math., Volume 23, Number 5 (2019), 1041-1059.

Dates
Received: 18 May 2018
Revised: 13 October 2018
Accepted: 11 November 2018
First available in Project Euclid: 21 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1542790915

Digital Object Identifier
doi:10.11650/tjm/181104

Mathematical Reviews number (MathSciNet)
MR4012368

Subjects
Primary: 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.)

Keywords
graph Laplacian matrix second largest Laplacian eigenvalue average degree Laplacian energy $\sigma$-invariant

Citation

Das, Kinkar Ch.; Mojallal, Seyed Ahmad. Open Problem on $\sigma$-invariant. Taiwanese J. Math. 23 (2019), no. 5, 1041--1059. doi:10.11650/tjm/181104. https://projecteuclid.org/euclid.twjm/1542790915


Export citation

References

  • W. N. Anderson and T. D. Morley, Eigenvalues of the Laplacian of a graph, Linear and Multilinear Algebra 18 (1985), no. 2, 141–145.
  • K. C. Das, The largest two Laplacian eigenvalues of a graph, Linear Multilinear Algebra 52 (2004), no. 6, 441–460.
  • ––––, A sharp upper bound for the number of spanning trees of a graph, Graphs Combin. 23 (2007), no. 6, 625–632.
  • ––––, A conjecture on algebraic connectivity of graphs, Taiwanese J. Math. 19 (2015), no. 5, 1317–1323.
  • K. C. Das, I. Gutman, A. S. Çevik and B. Zhou, On Laplacian energy, MATCH Commun. Math. Comput. Chem. 70 (2013), no. 2, 689–696.
  • K. C. Das and S. A. Mojallal, On Laplacian energy of graphs, Discrete Math. 325 (2014), 52–64.
  • K. C. Das, S. A. Mojallal and I. Gutman, On Laplacian energy in terms of graph invariants, Appl. Math. Comput. 268 (2015), 83–92.
  • K. C. Das, S. A. Mojallal and V. Trevisan, Distribution of Laplacian eigenvalues of graphs, Linear Algebra Appl. 508 (2016), 48–61.
  • K. Fan, On a theorem of Weyl concerning eigenvalues of linear transformations I, Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 652–655.
  • R. Grone, R. Meris and V. S. Sunder, The Laplacian spectrum of a graph, SIAM J. Matrix Anal. Appl. 11 (1990), no. 2, 218–238.
  • I. Gutman and B. Zhou, Laplacian energy of a graph, Linear Algebra Appl. 414 (2006), no. 1, 29–37.
  • S. T. Hedetniemi, D. P. Jacobs and V. Trevisan, Domination number and Laplacian eigenvalue distribution, European J. Combin. 53 (2016), 66–71.
  • J. v. d. Heuvel, Hamilton cycles and eigenvalues of graphs, Linear Algebra Appl. 226/228 (1995), 723–730.
  • J.-S. Li and Y.-L. Pan, A note on the second largest eigenvalue of the Laplacian matrix of a graph, Linear and Multilinear Algebra 48 (2000), no. 2, 117–121.
  • R. Merris, The number of eigenvalues greater than two in the Laplacian spectrum of a graph, Portugal. Math. 48 (1991), no. 3, 345–349.
  • ––––, Laplacian matrices of graphs: A survey, Linear Algebra Appl. 197/198 (1994), 143–176.
  • G. J. Ming and T. S. Wang, A relation between the matching number and Laplacian spectrum of a graph, Linear Algebra Appl. 325 (2001), no. 1-3, 71–74.
  • Y.-L. Pan and Y. P. Hou, Two necessary conditions for $\lambda_2(G) = d_2(G)$, Linear Multilinear Algebra 51 (2003), no. 1, 31–38.
  • S. Pirzada and H. A. Ganie, On the Laplacian eigenvalues of a graph and Laplacian energy, Linear Algebra Appl. 486 (2015), 454–468.
  • M. Robbiano and R. Jiménez, Applications of a theorem by Ky Fan in the theory of Laplacian energy of graphs, MATCH Commun. Math. Comput. Chem. 62 (2009), no. 3, 537–552.
  • J. R. Schott, Matrix Analysis for Statistics, Wiley Series in Probability and Statistics: Applied Probability and Statistics, John Wiley & Sons, New York, 1997.
  • Y. Wu, G. Yu and J. Shu, Graphs with small second largest Laplacian eigenvalue, European J. Combin. 36 (2014), 190–197.