**1. Introduction**

The distance Laplacian and distance signless Laplacian matrices of a graph *G* proposed by Aouchiche and Hansen [1] are defined as L(*G*) = *diag*(*Tr*) − D(*G*) and Q(*G*) = *diag*(*Tr*) + D(*G*), respectively. Much attention has been paid to them since they were put forward. Aouchiche et al. [2] described some elementary properties of the distance Laplacian eigenvalues of graphs. Niu et al. [3] determined some extremal graphs minimizing the distance Laplacian spectral radius among bipartite graphs in terms of the matching number and the vertex connectivity, respectively. Nath and Paul [4] focused on the graph whose complement is a tree or a unicyclic graph and considered the second-smallest distance Laplacian eigenvalue. Lin and Zhou [5] determined some extremal graphs among several classes of graphs. Tian et al. [6] proved four conjectures put forward by Aouchiche and Hansen in [2]. One can refer to [7–11] for more details on the distance signless Laplacian spectral radius of graphs.

Although lots of conclusions have been obtained, many more problems remain unsolved. For instance, there are few papers focusing on the distance (signless) Laplacian spectral radius of graphs in terms of diameter, an important parameter of graphs. For adjacency matrices of graphs, several conclusions with respect to the diameter have been derived (e.g., [12–14]). In [12], the authors determined some extremal graphs with small diameter. Generally, the communication network is organized with small diameter to improve the quality of the service on the networks. Motivated by this, in the present paper, we deduce a lower bound of the distance Laplacian spectral radius among bipartite graphs with diameter 4, and we hope that it could be used to address a general case.

This paper is arranged as follows. In Section 2, some elementary notions and lemmas applied in the next parts are presented. In Section 3, the lower bound for the distance Laplacian spectral radius is obtained for bipartite graphs with diameter 4. Moreover, the extremal graph attaining the lower bound is determined.

### **2. Preliminaries**

All graphs considered in this paper are undirected, connected and simple. By *V*(*G*), we denote the vertex set of *G*, and the order of *G* is |*V*(*G*)|. Denote by *NG*(*u*) the set of vertices adjacent to *u*. If *NG*(*u*) = *NG*(*v*) for *u*, *v* ∈ *V*(*G*), then they are called twin points. Generally, a subset *S* ⊂ *V*(*G*) is called a twin point set, if *NG*(*u*) = *NG*(*v*) for any *u*, *v* ∈ *S*. The distance between *u*, *v* ∈ *V*(*G*), denoted by *d*(*u*, *v*), is the length of the shortest path between *u* and *v*. The diameter of graph *G*, written as *d*(*G*) (*d* for short), is the maximum distance among all pairs of vertices of *G*. The chromatic number of *G* means the least number of colors required to color all the vertices of *G* such that each pair of adjacent vertices has different colors. The spanning subgraph of *G* is obtained by deleting some edges from *G* with order invariable. The transmission *TrG*(*u*) of a vertex *u* is referred to as the sum of the distances of *u* to all other vertices of *V*(*G*), i.e., *TrG*(*u*) = ∑*v*∈*V*(*G*) *d*(*v*, *u*). *Trmax*(*G*) means the maximal vertex transmission of *G*. Let B*n*,*<sup>d</sup>* be the set of all *n*-vertex bipartite graphs with diameter *d* and C*n*,*<sup>k</sup>* the set of all *n*-vertex graphs with chromatic number *k*.

Suppose *V*(*G*) = {*v*1, *v*2, ... , *vn*}. The distance matrix D(*G*) of *G* is an *n* × *n* symmetric real matrix with *d*(*vi*, *vj*) as the (*i*, *j*)-entry. Let the diagonal matrix *diag*(*Tr*), called the vertex transmission matrix of *G*, be

$$\operatorname{diag}(\operatorname{Tr}) = \operatorname{diag}(\operatorname{Tr}\_{\mathbb{G}}(\upsilon\_1), \operatorname{Tr}\_{\mathbb{G}}(\upsilon\_2), \dots, \operatorname{Tr}\_{\mathbb{G}}(\upsilon\_n)).$$

The largest eigenvalue of the distance Laplacian matrix L(*G*) is called the distance Laplacian spectral radius, written as *ρ*L(*G*). For any matrix *M*, *λ*1(*M*) always denotes the largest eigenvalue of *M*.

A vector *x* = (*x*1, *x*2, ... , *xn*)*<sup>T</sup>* can be considered as a function defined on *V*(*G*) = {*v*1, *v*2,..., *vn*}, which maps *vi* to *xi*, i.e., *x*(*vi*) = *xi*. Thus, for L(*G*),

$$\mathbf{x}^T \mathcal{L}(G)\mathbf{x} = \sum\_{\{\mathsf{u}, \mathsf{v}\} \subseteq V(G)} d(\mathsf{u}, \mathsf{v}) (\mathsf{x}(\mathsf{u}) - \mathsf{x}(\mathsf{v}))^2.$$

It is clear that **1** = (1, 1, ... , 1)*<sup>T</sup>* is an eigenvector corresponding to the eigenvalue zero of L(*G*). Thus, if *x* = (*x*1, *x*2, ... , *xn*)*<sup>T</sup>* is an eigenvector of L(*G*) corresponding to a nonzero eigenvalue, then ∑*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> *xi* = 0.

**Lemma 1** (Rayleigh's Principal Theorem, p. 29, [15])**.** *Let A be a symmetric real matrix and u any unit nonzero vector. Then λ*1(*A*) ≥ *u<sup>T</sup> A u with equality if and only if u is the eigenvector corresponding to λ*1(*A*)*.*

**Lemma 2** (Courant-Weyl Inequality, p. 31, [15])**.** *Let A*<sup>1</sup> *and A*<sup>2</sup> *be two symmetric real matrices of order n. Then*

$$
\lambda\_n(A\_2) + \lambda\_i(A\_1) \le \lambda\_i(A\_1 + A\_2) \text{ for } 1 \le i \le n.
$$

**Lemma 3** (Interlacing Theorem, p. 30, [15])**.** *Suppose A is a symmetric real matrix of order n and M a principal submatrix of A with order s*(≤ *n*)*. Then*

$$
\lambda\_i(A) \ge \lambda\_i(M), \ 1 \le i \le s.
$$

The next lemma follows from Lemma 3 immediately.

**Lemma 4** (Proposition 2.11, [6])**.** *Let G be an n-vertex graph and M a principal submatrix of* L(*G*) *with order s* ≤ *n. Then λ*1(*M*) ≤ *ρ*L(*G*)*.*

**Lemma 5** (Theorem 3.5, [1])**.** *Suppose G* + *euv is the graph obtained from G by adding an edge euv joining u and v. Then ρ*L(*G*) ≥ *ρ*L(*G* + *uv*).
