*Article* **A Lower Bound for the Distance Laplacian Spectral Radius of Bipartite Graphs with Given Diameter**

**Linming Qi 1,\*, Lianying Miao 2, Weiliang Zhao <sup>1</sup> and Lu Liu <sup>3</sup>**


**Abstract:** Let *G* be a connected, undirected and simple graph. The distance Laplacian matrix L(*G*) is defined as L(*G*) = *diag*(*Tr*) − D(*G*), where D(*G*) denotes the distance matrix of *G* and *diag*(*Tr*) denotes a diagonal matrix of the vertex transmissions. Denote by *ρ*L(*G*) the distance Laplacian spectral radius of *G*. In this paper, we determine a lower bound of the distance Laplacian spectral radius of the *n*-vertex bipartite graphs with diameter 4. We characterize the extremal graphs attaining this lower bound.

**Keywords:** distance Laplacian matrix; spectral radius; diameter

**MSC:** 05C50

**Citation:** Qi, L.; Miao, L.; Zhao, W.; Liu, L. A Lower Bound for the Distance Laplacian Spectral Radius of Bipartite Graphs with Given Diameter. *Mathematics* **2022**, *10*, 1301. https://doi.org/10.3390/ math10081301

Academic Editors: Irina Cristea and Hashem Bordbar

Received: 16 March 2022 Accepted: 12 April 2022 Published: 14 April 2022

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