Abstract
The distance Laplacian matrix of a connected graph is defined as , where is the distance matrix of and is the diagonal matrix of vertex transmissions of . The largest eigenvalue of is called the distance Laplacian spectral radius of . In this paper, we determine the graphs with maximum and minimum distance Laplacian spectral radius among all clique trees with vertices and cliques. Moreover, we obtain vertices and cliques.
1. Introduction
In this paper, we consider simple connected graphs [1]. A graph G is represented by , in which the set represents its vertex set and is the edge set connecting pairs of distinct vertices. The number is referred to as the order of . The distance matrix of is the matrix , where denotes the distance between vertices and in , i.e., the length of a shortest path from to in . For , the transmission of in , denoted by , is defined as the sum of distances from to all other vertices of . Let be the diagonal matrix of vertex transmissions of . In 2013, Aouchiche and Hansen [2] first gave the definition of distance Laplacian matrix: for a connected graph , , where denotes the distance Laplacian matrix. Obviously, is a positive semidefinite, symmetric, and singular matrix. The distance Laplacian eigenvalues of , denoted by , are the eigenvalues of . Especially, the largest eigenvalue is the distance Laplacian spectral radius of . The positive unit eigenvector, i.e., all components of the eigenvector are positive, corresponding to is called the Perron eigenvector of .
For a graph , two vertices are called adjacent if they are connected by an edge and two edges are called incident if they share a common vertex. The set of vertices that are adjacent to a vertex is called the neighborhood of and is presented by . As usual, let , , and denote the complete graph, the star, and the path with order , respectively. is a connected graph, , is not connected, and then is a cut-vertex set. If has only vertex , then is a cut-vertex. A block of is a maximal connected subgraph of that has no cut-vertex. A block is a clique if the block is a complete graph. A graph is a clique tree if each block of is a clique. We call a clique path if we replace each edge of by a clique such that for and for and . We call a clique star if we replace each edge of the star with a clique such that for and (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 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 and in the form of , , 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 , 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.
The distance Laplacian energy is defined as , where is the average transmission of and is defined by . 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. The 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 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 but also get their distance Laplacian energies; second, we prove all clique stars are the graphs with minimum distance Laplacian spectral radius among all clique trees with vertices and cliques. Then, we show that the clique path for is the graph with maximum distance Laplacian spectral radius among all clique trees with vertices and cliques.
2. Preliminaries
Let be a connected graph with . A column vector can be considered as a function defined on which maps vertex to , i.e., for . Then,and is a distance Laplacian eigenvalue with corresponding eigenvector if and only if , for each ,or equivalently
The above equation is called the eigenequation of at .
Note that is an eigenvector of corresponding to . For , if is an eigenvector of corresponding to , we have .
For a unit column vector , by Rayleigh’s principle, we have with equality if and only if is an eigenvector of corresponding to .
The following is the well-known Cauchy interlacing theorem.
Lemma 1 (Cauchy interlace theorem) (see [1]). Let be a Hermitian matrix with eigenvalues and be one of its principal submatrices. Let have eigenvalues . Then, the inequalities hold.
Lemma 2 (see [6]). Let be a connected graph with three induced subgraphs , , and such that for and for and (see Figure 2). For and , let and . If , then or .
3. Minimum Distance Laplacian Spectral Radius of Clique Trees
The diameter of a graph is the maximum distance between any pair of vertices.
Lemma 3. Let be a clique tree with vertices and cliques. If , then .
Proof. For convenience, let and be a clique path of . Denote the cliques of by , . Let for . Let and . Then, is a diameter path of . We can easily getThen, we have Let be the principal submatrix of indexed by and . Then,and thus By Lemma 2, we have .
Theorem 1. Let be an arbitrary clique star with vertices and cliques. Then, .
Proof. Obviously, we have . Let be a Perron eigenvector of corresponding to . By symmetry, we may assume for any , . Let , then we have Thus, is the largest root of the equation , where and Therefore, we have n and 0 are also distance Laplacian eigenvalues of .
Combining Lemma 3 and Theorem 1, we have the following result.
Theorem 2. Among all clique trees with vertices and cliques, the graphs attaining the minimum distance Laplacian spectral radius are clique stars .
Let be the identity matrix of order . The characteristic polynomial of can be written as . Let us label the vertices of such that is the first vertices, and the first vertices are from , the following vertices are from , …, and the last are from . Let . Combining Theorem 1, by direct calculations, we get the following result.
Corollary 1. The distance Laplacian eigenvalues of are of multiplicities , of multiplicities , , and 0.
Theorem 3. Let be an arbitrary clique star with vertices and cliques. Then, we have .
Proof. Obviously, we have . For convenience, let . For any , we have . Let , . Then, we have and . By Cauchy–Schwarz inequality, we have . So, we get . By Corollary 1, we know for , and . since is equal to the trail of , i.e., . So, we get .
4. Maximum Distance Laplacian Spectral Radius of Clique Trees
Lemma 4. Let be a connected graph and be a clique tree with . Suppose is a clique path of with cliques , , and for . Let be the graph obtained by identifying a vertex of and a vertex of , where . Then, or .
Proof. By Lemma 2, we may assume or for . Denote the component of which contains vertex by and the component of which contains vertex by . Let , , and . Suppose is a Perron eigenvector of corresponding to . In the following, we will first prove or . Case 1: . From to , we have Thus, . In the following, we will prove . If , then , which implies for any , , and is also a Perron eigenvector of corresponding to . For arbitrary , from the eigenequations of and at , we have So, we have . Similarly, for arbitrary and , we have and . Then, we have for any . Since , we have , which implies and . From the eigenequation of at and , we have , which is a contradiction. Up to now, we have proved . Case 2: .From to , we have Then, we have Thus, .
In the following, we will prove or .
If , we may denote the component of which contains vertex by and the component of which contains vertex by . Let , , and . Let be a Perron eigenvector of corresponding to . If , then , and we can get , which is a contradiction. So, we have . Then, we have , 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 .
Similarly, if , we can prove .
Theorem 4. Among all clique trees with vertices and cliques, the graph attaining the maximum distance Laplacian spectral radius is for some .
Proof. Let be the graph with maximum distance Laplacian spectral radius among all clique trees with vertices and cliques. By Lemma 4, we get . Let for . If , the result holds. Next, we may assume . Suppose there exists some such that . Denote the component of which contains vertex by and the component of which contains vertex by . Let , , and , i.e., . Let i.e., and . Suppose is a Perron eigenvector of corresponding to . In the following, we will first prove or . Case 1: . From to , we have which implies . Similar to Case 1 of Lemma 4, we can get the equal sign in the above inequality does not hold. So, we have . Case 2: . Then, we have Thus, we have .Doing the above graph transformations until , we get as for some .
5. Conclusion
This paper mainly determines the extremal graphs with maximum and minimum distance Laplacian spectral radius among all clique trees with vertices and cliques. Moreover, we get the distance Laplacian energies of all the clique stars with vertices and cliques. Based on our results, we conjecture that the line graphs of and are the unique graphs with minimum and maximum distance Laplacian spectral radius among all the line graphs of unicyclic graphs, respectively, where is the graph obtained by adding an edge to the star of order and is the graph obtained by adding an edge between a vertex of a triangle and a terminal vertex of a path on 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
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the NSFC (grant nos. 11501491, 11671347, and 61771019).