The Distance Laplacian Spectral Radius of Clique Trees

-e distance Laplacian matrix of a connected graph G is defined asL(G) � Tr(G) − D(G), where D(G) is the distance matrix of G and Tr(G) is the diagonal matrix of vertex transmissions of G. -e largest eigenvalue ofL(G) is called the distance Laplacian spectral radius of G. In this paper, we determine the graphs with maximum and minimum distance Laplacian spectral radius among all clique trees with n vertices and k cliques. Moreover, we obtainn vertices and k cliques.


Introduction
In this paper, we consider simple connected graphs [1]. A graph G is represented by G � (V(G), E(G)), in which the set V(G) � v 1 , v 2 , . . . , v n represents its vertex set and E(G) is the edge set connecting pairs of distinct vertices. e number n � |V(G)| is referred to as the order of G. e distance matrix of G is the n × n matrix D(G) � (d G (u, v)) u,v∈V(G) , where d G (u, v) denotes the distance between vertices u and v in G, i.e., the length of a shortest path from u to v in G. For u ∈ V(G), the transmission of u in G, denoted by Tr G (u), is defined as the sum of distances from u to all other vertices of G. Let Tr(G) be the diagonal matrix of vertex transmissions of G. In 2013, Aouchiche and Hansen [2] first gave the definition of distance Laplacian matrix: for a connected graph G, L(G) � Tr(G) − D(G), where L(G) denotes the distance Laplacian matrix. Obviously, L(G) is a positive semidefinite, symmetric, and singular matrix. e distance Laplacian eigenvalues of G, denoted by λ 1 (G) ≥ λ 2 (G) ≥ · · · ≥ λ n (G) � 0, are the eigenvalues of L(G). Especially, the largest eigenvalue λ 1 (G) is the distance Laplacian spectral radius of G. e positive unit eigenvector, i.e., all components of the eigenvector are positive, corresponding to λ 1 (G) is called the Perron eigenvector of L(G).
For a graph G, two vertices are called adjacent if they are connected by an edge and two edges are called incident if they share a common vertex. e set of vertices that are adjacent to a vertex v ∈ V(G) is called the neighborhood of v and is presented by N G (v). As usual, let K n , K 1,n− 1 , and P n denote the complete graph, the star, and the path with order n, respectively. G is a connected graph, X ∈ V(G), G − X is not connected, and then X is a cut-vertex set. If X has only vertex v, then v is a cut-vertex. A block of G is a maximal connected subgraph of G that has no cut-vertex. A block is a clique if the block is a complete graph. A graph G is a clique tree if each block of G is a clique. We call P n 1 ,n 2 ,...,n k a clique path if we replace each edge of P k+1 by a clique K n i such that We call K u,n 1 ,n 2 ,...,n k a clique star if we replace each edge of the star K 1,k with a clique K n i such that V(K n i ) ∩ V(K n j ) � u for i ≠ j and i, j � 1, 2, . . . , k (see Figure 1).
Recently, Xing and Zhou [3] characterized the unique graph with minimum distance Laplacian spectral radius among all the bicyclic graphs with fixed number of vertices; Aouchiche and Hansen [4] showed that the star K 1,n is the unique tree with the minimum distance Laplacian spectral radius among all trees; Lin et al. [5,6] determined the unique graph with minimum distance Laplacian spectral radius among all the trees with fixed bipartition, nonstar-like trees, noncaterpillar trees, nonstar-like noncaterpillar trees, and the graph with fixed edge connectivity at most half of the order, respectively; Niu et al. [7] determined the unique graph with minimum distance Laplacian spectral radius among all the bipartite graphs with fixed matching number and fixed vertex connectivity, respectively; Fan et al. [8] determined the graph with minimum distance Laplacian spectral radius among all the unicyclic and bicyclic graphs with fixed numbers of vertices, respectively; Lin and Zhou [9] determined the unique graph with maximum distance Laplacian spectral radius among all the unicyclic graphs with fixed numbers of vertices.
In 2019, Cui et al. [10] investigated a convex combination of Tr(G) and which is called the generalized distance matrix. Alhevaz et al. [11] gave some new upper and lower bounds for the generalized distance energy of graphs which are established based on parameters including the Wiener index and the transmission degrees and found that the complete graph has the minimum generalized distance energy among all connected graphs; Lin and Drury et al. [12] established some bounds for the generalized distance Gaussian Estrada index of a connected graph, involving the different graph parameters, including the order, the Wiener index, the transmission degrees, and the parameter α ∈ [0, 1], and characterized the extremal graphs attaining these bounds; Alhevaz et al. [13] obtained some bounds for the generalized distance spectral radius of graphs using graph parameters like the diameter, the order, the minimum degree, the second minimum degree, the transmission degree, and the second transmission degree and characterized the extremal graphs; Alhevaz et al. [14] studied the generalized distance spectrum of join of two regular graphs and join of a regular graph with the union of two different regular graphs; Shang [15] established better lower and upper bounds to the distance Estrada index for almost all graphs. e distance Laplacian energy is defined as Although there has been extensive work done on the distance Laplacian spectral radius of graphs, relatively little is known in regard to distance Laplacian energy. e distance Laplacian energy was first introduced in [16], where several lower and upper bounds were obtained; Das et al. [17] gave some lower bounds on distance Laplacian energy in terms of n for graphs and trees and characterized the extremal graphs and trees. In this paper, first, we not only get the distance Laplacian eigenvalues of all clique stars K u,n 1 ,n 2 ,...,n k but also get their distance Laplacian energies; second, we prove all clique stars K u,n 1 ,n 2 ,...,n k are the graphs with minimum distance Laplacian spectral radius among all clique trees with n vertices and k cliques. en, we show that the clique path P m,2,...,2,n− m− k+3 for m ≥ 3 is the graph with maximum distance Laplacian spectral radius among all clique trees with n vertices and k cliques.

Preliminaries
and λ is a distance Laplacian eigenvalue with corresponding eigenvector x if and only if x ≠ 0, for each u ∈ V(G), or equivalently e above equation is called the eigenequation of G at u.
For a unit column vector x ∈ R n , by Rayleigh's principle, e following is the well-known Cauchy interlacing theorem.
Lemma 1 (Cauchy interlace theorem) (see [1]). Let A be a Hermitian matrix with eigenvalues λ 1 ≥ · · · ≥ λ n and B be one of its principal submatrices. Let B have eigenvalues Lemma 2 (see [6]). Let G be a connected graph with three induced subgraphs G 1 , 2 Discrete Dynamics in Nature and Society

Minimum Distance Laplacian Spectral
Radius of Clique Trees e diameter of a graph is the maximum distance between any pair of vertices. Lemma 3. Let S be a clique tree with n vertices and k cliques.
Proof. For convenience, let diam(S) � d and P n 1 ,n 2 ,...,n d be a clique path of S. Denote the cliques of P n 1 ,n 2 ,...,n d by K n 1 , We can easily get en, we have Let M be the principal submatrix of L(S) indexed by v 0 and v d . en, and thus By Lemma 2, we have λ 1 (S) ≥ λ 1 (M) > 2n − 1.
Combining Lemma 3 and eorem 1, we have the following result. □ Theorem 2. Among all clique trees with n vertices and k cliques, the graphs attaining the minimum distance Laplacian spectral radius are clique stars K u,n 1 ,n 2 ,...,n k .
Let I be the identity matrix of order n. e characteristic polynomial of L(G) can be written as ψ(G: λ) � det(λI − L(G)). Let us label the vertices of K u,n 1 ,n 2 ,...,n k such that u is the first vertices, and the first n 1 vertices are from V(K n 1 ), the following n 2 − 1 vertices are from V(K n 2 )\ u { }, . . ., and the last n k − 1 are from V(K n k )\ u { }. Let det(λI − L(K u,n 1 ,n 2 ,...,n k )) � 0. Combining eorem 1, by direct calculations, we get the following result. 4 Discrete Dynamics in Nature and Society Corollary 1.

Maximum Distance Laplacian Spectral Radius of Clique Trees
Lemma 4. Let H be a connected graph and S be a clique tree with diam(S) � d. Suppose P n 1 ,n 2 ,...,n d is a clique path of S with cliques K n 1 , K n 2 , . . ., K n d and Proof. By Lemma 2, we may assume u . Suppose x is a Perron eigenvector of L(H t ) corresponding to λ 1 (H t ). In the following, we will first prove us, λ 1 (H t+1 ) ≥ λ 1 (H t ). In the following, we will prove , and x is also a Perron eigenvector of L(H t+1 ) corresponding to λ 1 (H t+1 ). For arbitrary ω 1 ∈ S 1 , from the eigenequations of H t+1 and H t at ω 1 , we have Discrete Dynamics in Nature and Society Similarly, for arbitrary ω 2 ∈ S 2 and ω 3 ∈ S 3 \ v t , we have From the eigenequation of H t at v 1 and v 2 , we have which is a contradiction. Up to now, we have proved λ 1 (H t+1 ) > λ 1 (H t ).
From H t to H t− 1 , we have en, we have us, λ 1 (H t− 1 ) > λ 1 (H t ). In the following, we will prove , and we can get λ 1 (H t ) > λ 1 (H t+1 ), which is a contradiction. So, we have ≥ 0, similar to case 1, and we can get the equal sign in the above inequality does not hold. So, we have Repeating the above procedure, we can get □ Theorem 4. Among all clique trees with n vertices and k cliques, the graph attaining the maximum distance Laplacian spectral radius is P m,2,...,2,n− m− k+3 for some m ≥ 3.

Conclusion
is paper mainly determines the extremal graphs with maximum and minimum distance Laplacian spectral radius among all clique trees with n vertices and k cliques. Moreover, we get the distance Laplacian energies of all the clique stars with n vertices and k cliques. Based on our results, we conjecture that the line graphs of S + n and K i n ,3 are the unique graphs with minimum and maximum distance Laplacian spectral radius among all the line graphs of unicyclic graphs, respectively, where S + n is the graph obtained by adding an edge to the star K 1,n− 1 of order n and K i n ,3 is the graph obtained by adding an edge between a vertex of a triangle and a terminal vertex of a path on n − 3 vertices. Moreover, we can study the distance Laplacian spectral radius of diclique trees in the future.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.