Taiwanese Journal of Mathematics

Open Problem on $\sigma$-invariant

Kinkar Ch. Das and Seyed Ahmad Mojallal

Advance publication

This article is in its final form and can be cited using the date of online publication and the DOI.

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., Advance publication (2019), 19 pages.

Dates
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

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., advance publication, 21 November 2018. 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.