The Upper Bound For The Largest Signless Laplacian Eigenvalue Of Weighted Graphs

Abstract

In this study, we find an upper bound for the largest signless Laplacian eigenvalue of simple connected weighted graphs, where the edge weights are positive definite square matrices. Also we obtain some results on weighted and unweighted graphs by using this bound.

Keywords

• Weighted graph
• signless Laplacian eigenvalue
• upper bound

References

1. REFERENCES
2. Anderson, W.N. and Morley, T.D., “Eigenvalues Of The Laplacian Of A Graph”, Linear and Multilinear Algebra, 18(2): 141-145, (1985).
3. Das, K.C. and Bapat, R.B., “A Sharp Upper Bound On The Largest Laplacian Eigenvalue Of Weighted Graphs”, Linear Algebra and its Applications, 409: 153-165, (2005).
4. Das, K.C., “Extremal Graph Characterization From The Upper Bound Of The Laplacian Spectral Radius Of Weighted Graphs”, Linear Algebra and its Applications, 427(1): 55-69, (2007).
5. Das, K.C. and Bapat, R.B., “A Sharp Upper Bound On The Spectral Radius Of Weighted Graphs”, Discrete Mathematics, 308(15): 3180-3186, (2008).
6. Horn, R.A. and Johnson, C.R., “Matrix Analysis”, 2 nd ed., Cambridge/United Kingdom:Cambridge University Press, 225-260, 391-425, (2012).
7. Maden, A.D., Das, K.C. and Çevik, A.S., “Sharp Upper Bounds On The Spectral Radius Of The Signless Laplacian Matrix Of A Graph”, Applied Mathematics and Computation, 219(10): 5025-5032, (2013).
8. Sorgun, S. and Büyükköse, Ş., “On The Bounds For The Largest Laplacian Eigenvalues Of Weighted Graphs”, Discrete Optimization, 9(2): 122-129, (2012).

9. Sorgun, S., Büyükköse, Ş. and Özarslan, H.S., “An Upper Bound On The Spectral Radius Of Weighted Graphs”, Hacettepe Journal of Mathematics and Statistics, 42(5): 517-524, (2013).
10. Zhang, F., “Matrix Theory: Basic Results And Techniques”, 1 nd ed., New York/USA:Springer-Verlag, 159-173, (1999).