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Improved Bounds for the Extremal Non-Trivial Laplacian Eigenvalues

Year 2015, Volume: 28 Issue: 1, 65 - 68, 23.02.2015

Abstract

Let G be a simple connected graph and its Laplacian eigenvalues be µ1≥ µ2≥…≥ µn-1≥ µn=0. In this paper, we present an upper bound for the algebraic connectivity µn-1 of G and a lower bound for the largest eigenvalue µ1 of G in terms of the degree sequence d1,d2,…,dn of G and the number Ni∩Nj of common vertices of i and j (1≤i<j≤n) and hence we improve bounds of Maden and Büyükköse [14].

References

  • Anderson W.N., Morley T.D., Eigenvalues of the Laplacian of a graph, Linear Multilinear Algebra 18 (1985) 141-145.
  • Büyükköse Ş., Bounds for singular values using matrix traces, Msc Thesis, Selçuk University, (2000).
  • Chung F.R.K., Eigenvalues of Graphs, In Proceeding of the International Congress of Mathematician, 1333-1342. Switzerland, (1994)
  • Das K. Ch., An improved upper bound for Laplacian graph eigenvalues, Linear Algebra Appl. 368 (2003) 269-278.
  • Fiedler M., Algebraic connectivity of graphs, Czechoslovak Math. J. 23 (1973) 298-305.
  • Li J.-S., Zhang D., A new upper bound for eigenvalues of the Laplacian matrix of a graph, Linear Algebra Appl. 265 (1997) 93-100.
  • Li J.-S., Zhang D., On Laplacian eigenvalues of a graph, Linear Algebra Appl. 285 (1998) 305-307.
  • Lu M., Zhang L., Tian F., Lower bounds of the Laplacian spectrum of graphs based on diameter, Linear Algebra Appl. 420 (2007) 400-406.
  • Merris R., A note on Laplacian graph eigenvalues, Linear Algebra Appl. 285 (1998) 33-35.
  • Merris R., Laplacian matrices of Graphs: a Survey, Linear Algebra and its Applications, 1 97, 198(1994) 143-176
  • Mohar B., The Laplacian spectrum of graphs, In Graph Theory, Combinatorics and Applications, Vol.2 (1998) 871-898.
  • Rojo O., Soto R., Rojo H., An always nontrivial upper bound for Laplacian graph eigenvalues, Linear Algebra Appl. 312(2000),155-159.
  • Wolkowicz H., Styan G.P.H., Bounds for eigenvalues using traces, Linear Algebra Appl. 29 (1980) 471-506.
  • Maden A.D., Büyükköse Ş., Bounds for Laplacian Graph Eigenvalues, Mathematical Inequalities and Applications, Vol 12, Num.3 (2012), 529-536.
  • Grone R., Merris R., The Laplacian spectrum of a graph, SIAM J. Discr. Math. 7 (1994) 221–229.
  • Li J.S., Pan Y.L., A note on the second largest eigenvalue of the Laplacian matrix of a graph, Linear Multilinear Algebra 48 (2000) 117–121.
Year 2015, Volume: 28 Issue: 1, 65 - 68, 23.02.2015

Abstract

References

  • Anderson W.N., Morley T.D., Eigenvalues of the Laplacian of a graph, Linear Multilinear Algebra 18 (1985) 141-145.
  • Büyükköse Ş., Bounds for singular values using matrix traces, Msc Thesis, Selçuk University, (2000).
  • Chung F.R.K., Eigenvalues of Graphs, In Proceeding of the International Congress of Mathematician, 1333-1342. Switzerland, (1994)
  • Das K. Ch., An improved upper bound for Laplacian graph eigenvalues, Linear Algebra Appl. 368 (2003) 269-278.
  • Fiedler M., Algebraic connectivity of graphs, Czechoslovak Math. J. 23 (1973) 298-305.
  • Li J.-S., Zhang D., A new upper bound for eigenvalues of the Laplacian matrix of a graph, Linear Algebra Appl. 265 (1997) 93-100.
  • Li J.-S., Zhang D., On Laplacian eigenvalues of a graph, Linear Algebra Appl. 285 (1998) 305-307.
  • Lu M., Zhang L., Tian F., Lower bounds of the Laplacian spectrum of graphs based on diameter, Linear Algebra Appl. 420 (2007) 400-406.
  • Merris R., A note on Laplacian graph eigenvalues, Linear Algebra Appl. 285 (1998) 33-35.
  • Merris R., Laplacian matrices of Graphs: a Survey, Linear Algebra and its Applications, 1 97, 198(1994) 143-176
  • Mohar B., The Laplacian spectrum of graphs, In Graph Theory, Combinatorics and Applications, Vol.2 (1998) 871-898.
  • Rojo O., Soto R., Rojo H., An always nontrivial upper bound for Laplacian graph eigenvalues, Linear Algebra Appl. 312(2000),155-159.
  • Wolkowicz H., Styan G.P.H., Bounds for eigenvalues using traces, Linear Algebra Appl. 29 (1980) 471-506.
  • Maden A.D., Büyükköse Ş., Bounds for Laplacian Graph Eigenvalues, Mathematical Inequalities and Applications, Vol 12, Num.3 (2012), 529-536.
  • Grone R., Merris R., The Laplacian spectrum of a graph, SIAM J. Discr. Math. 7 (1994) 221–229.
  • Li J.S., Pan Y.L., A note on the second largest eigenvalue of the Laplacian matrix of a graph, Linear Multilinear Algebra 48 (2000) 117–121.
There are 16 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Mathematics
Authors

Şerife Büyükköse

Ercan Altınışık This is me

Feyza Yalçın This is me

Publication Date February 23, 2015
Published in Issue Year 2015 Volume: 28 Issue: 1

Cite

APA Büyükköse, Ş., Altınışık, E., & Yalçın, F. (2015). Improved Bounds for the Extremal Non-Trivial Laplacian Eigenvalues. Gazi University Journal of Science, 28(1), 65-68.
AMA Büyükköse Ş, Altınışık E, Yalçın F. Improved Bounds for the Extremal Non-Trivial Laplacian Eigenvalues. Gazi University Journal of Science. February 2015;28(1):65-68.
Chicago Büyükköse, Şerife, Ercan Altınışık, and Feyza Yalçın. “Improved Bounds for the Extremal Non-Trivial Laplacian Eigenvalues”. Gazi University Journal of Science 28, no. 1 (February 2015): 65-68.
EndNote Büyükköse Ş, Altınışık E, Yalçın F (February 1, 2015) Improved Bounds for the Extremal Non-Trivial Laplacian Eigenvalues. Gazi University Journal of Science 28 1 65–68.
IEEE Ş. Büyükköse, E. Altınışık, and F. Yalçın, “Improved Bounds for the Extremal Non-Trivial Laplacian Eigenvalues”, Gazi University Journal of Science, vol. 28, no. 1, pp. 65–68, 2015.
ISNAD Büyükköse, Şerife et al. “Improved Bounds for the Extremal Non-Trivial Laplacian Eigenvalues”. Gazi University Journal of Science 28/1 (February 2015), 65-68.
JAMA Büyükköse Ş, Altınışık E, Yalçın F. Improved Bounds for the Extremal Non-Trivial Laplacian Eigenvalues. Gazi University Journal of Science. 2015;28:65–68.
MLA Büyükköse, Şerife et al. “Improved Bounds for the Extremal Non-Trivial Laplacian Eigenvalues”. Gazi University Journal of Science, vol. 28, no. 1, 2015, pp. 65-68.
Vancouver Büyükköse Ş, Altınışık E, Yalçın F. Improved Bounds for the Extremal Non-Trivial Laplacian Eigenvalues. Gazi University Journal of Science. 2015;28(1):65-8.