On the distance Laplacian spectral radius of bicyclic graphs

Abstract

The distance Laplacian matrix of a connected graph G is defined as $\mathcal{L}\left(G\right)=Tr\left(G\right)-D\left(G\right)$, where Tr(G) is the diagonal matrix of the vertex transmissions in G and D(G) is the distance matrix of G. The largest eigenvalue of $\mathcal{L}\left(G\right)$ is called the distance Laplacian spectral radius of G. In this paper, we determine the graphs with the maximum distance Laplacian spectral radius and the minimum distance Laplacian spectral radius among all the bicyclic graphs with given order, respectively.

Keywords:

• Distance Laplacian matrix
• eigenvalues
• spectral radius
• bicyclic graph

AMS Classifications:

• 05C50
• 15A18

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research is partially supported by the National Natural Science Foundation of China (Nos. 11771039,11971054).