Home > ELA > Vol. 31 (2016)

#### Article Title

#### Keywords

Distance Laplacian matrix, Laplacian matrix, largest eigenvalue, multiplicity of eigenvalues

#### Abstract

In this note, the graphs of order n having the largest distance Laplacian eigenvalue of multiplicity n −2 are characterized. In particular, it is shown that if the largest eigenvalue of the distance Laplacian matrix of a connected graph G of order n has multiplicity n − 2, then G = S_n or G = K_(p,p), where n = 2p. This resolves a conjecture proposed by M. Aouchiche and P. Hansen in [M. Aouchiche and P. Hansen. A Laplacian for the distance matrix of a graph. Czechoslovak Mathematical Journal, 64(3):751–761, 2014.]. Moreover, it is proved that if G has P_5 as an induced subgraph then the multiplicity of the largest eigenvalue of the distance Laplacian matrix of G is less than n − 3.

#### Recommended Citation

Marques da Silva, Celso Junior; Aguieiras Alvarez de Freitas, Maria; and Raposo Del-Vecchio, Renata.
(2016),
"A note on a conjecture for the distance Laplacian matrix",
*Electronic Journal of Linear Algebra*,
Volume 31, pp. 60-68.

DOI: http://dx.doi.org/10.13001/1081-3810.3002

*Abstract*