ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

Discussiones Mathematicae Graph Theory

IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

Rejection Rate (2017-2018): c. 84%

Discussiones Mathematicae Graph Theory


Discussiones Mathematicae Graph Theory 26(1) (2006) 49-58
DOI: 10.7151/dmgt.1300


Yi Wang and Yi-Zheng Fan

School of Mathematics and Computational Science
Anhui University, Hefei, Anhui 230039, P.R. China


In this paper, we determine all trees with the property that adding a particular edge will result in exactly two Laplacian eigenvalues increasing respectively by 1 and the other Laplacian eigenvalues remaining fixed. We also investigate a situation in which the algebraic connectivity is one of the changed eigenvalues.

Keywords: tree, Laplacian eigenvalues, spectral integral variation, algebraic connectivity.

2000 Mathematics Subject Classification: 05C50, 15A18.


[1] D.M. Cvetković, M. Doob and H. Sachs, Spectra of Graphs-Theory and Applications (2nd Edn., VEB Deutscher Verlag d. Wiss., Berlin, 1982).
[2] Yi-Zheng Fan, On spectral integral variations of graph, Linear and Multilinear Algebra 50 (2002) 133-142, doi: 10.1080/03081080290019513.
[3] Yi-Zheng Fan, Spectral integral variations of degree maximal graphs, Linear and Multilinear Algebra 52 (2003) 147-154.
[4] M. Fiedler, Algebraic connectivity of graphs, Czechoslovak Math. J. 23 (1973) 298-305.
[5] M. Fiedler, A property of eigenvectors of nonnegative symmetric matrices and its applications to graph theory, Czechoslovak Math. J. 25 (1975) 619-633.
[6] R. Grone, R. Merris and V.S. Sunder, The Laplacian spectrum of a graph, SIAM J. Matrix Anal. Appl. 11 (1990) 218-238, doi: 10.1137/0611016.
[7] R. Grone and R. Merris, The Laplacian spectrum of a graph II, SIAM J. Discrete Math. 7 (1994) 229-237, doi: 10.1137/S0895480191222653.
[8] F. Harary and A.J. Schwenk, Which graphs have integral spectra? in: Graphs and Combinatorics, R.A. Bari and F. Harray eds. (Springer-Verlag, 1974), 45-51, doi: 10.1007/BFb0066434.
[9] S. Kirkland, A characterization of spectrum integral variation in two places for Laplacian matrices, Linear and Multilinear Algebra 52 (2004) 79-98, doi: 10.1080/0308108031000122506.
[10] R. Merris, Laplacian matrices of graphs: a survey, Linear Algebra Appl. 197/198 (1994) 143-176, doi: 10.1016/0024-3795(94)90486-3.
[11] R. Merris, Degree maximal graphs are Laplacian integral, Linear Algebra Appl. 199 (1994) 381-389, doi: 10.1016/0024-3795(94)90361-1.
[12] B. Mohar, The Laplacian spectrum of graphs, in: Y. Alavi et al. (eds.), Graph Theory, Combinatorics, and Applications (Wiley, New York, 1991) 871-898.
[13] W. So, Rank one perturbation and its application to the Laplacian spectrum of graphs, Linear and Multilinear Algebra 46 (1999) 193-198, doi: 10.1080/03081089908818613.

Received 11 October 2004
Revised 8 January 2005