A NOTE ON A CONJECTURE FOR THE DISTANCE LAPLACIAN MATRIX

作者：da Silva Celso M Jr^*; de Freitas Maria Aguieiras A; Del Vecchio Renata R

来源：Electronic Journal of Linear Algebra, 2016, 31(1): 60-68.

DOI：10.13001/1081-3810.3002

• Distance Laplacian matrix
• Laplacian matrix
• Largest eigenvalue
• Multiplicity of eigenvalues

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 congruent to S-n or G congruent to K-p,K-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):751761, 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.

• 出版日期2016-2