Abstract
Let G be a graph and denote by Q(G)=D(G)+A(G), L(G)=D(G)−A(G) the sum and the difference between the diagonal matrix of vertex degrees and the adjacency matrix of G, respectively. In this paper, some properties of the matrix Q(G) are studied. At the same time, a necessary and sufficient condition for the equality of the spectrum of Q(G) and L(G) is given.
Similar content being viewed by others
References
Merris, R., Laplacian matrices of graphs: A survey, Linear Algebra Appl., 1994(197):143–176.
Mohar, B., The Laplacian spectrum of graphs, In: Alavi, Y., Chartrand, G., Oellermann, O. R., et al. eds., Proceedings of the Sixth Quadrennial International Conference on the Theory and Applications of Graphs, Western Michigan University, Kalamazoo, Wiley, New York, 1991,871–898.
Cvetkovic, D. M., Boob, M. and Sachs, H., Spectra of Graph-Theory and Applications, Academic Press, New York, 1980.
Desai, M. and Rao, V., A characterization of the smallest eigenvalue of a graph, Journal of Graph Theory, 1994, 18(2):181–194.
Van Den Heuvel, J., Hamilton cycles and eigenvalues of graphs, Linear Algebra Appl., 1995(226–228): 723–730.
Biggs, N.L., Algebraic Graph Theory, Cambridge University Press, 1974.
Minc, H., Nonnegative Matrices, John Wiley and Sons, New York, 1988.
Horn, R.A. and Johnson, C.R., Matrix Analysis, Cambridge University Press, 1985.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
An, C. Some properties of the spectrum of graphs. Appl. Math. Chin. Univ. 14, 103–107 (1999). https://doi.org/10.1007/s11766-999-0061-7
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s11766-999-0061-7