Abstract

Let be a simple connected graph with vertex set and be the degree of the vertex . Let be the distance matrix and be the diagonal matrix of the vertex transmissions of . The generalized distance matrix of is defined as , where . If are the eigenvalues of , then the generalized distance spectral radius of is defined as . The generalized distance energy of is , where is the Wiener index of . In this paper, we give some bounds of the generalized distance spectral radius and the generalized distance energy.

1. Introduction

Throughout this paper, we consider simple, connected, and finite graphs. Let be such graph with vertex set and edge set . Let denote the degree of vertex and denote the neighbor set of . The distance between vertices and in is the length of the shortest path connecting to , which is denoted as . The distance matrix [1, 2] of is an matrix , where for .

Definition 1. Let be a graph with vertex set . The transmission of vertex , denoted by or , is defined to be the sum of the distances from to all vertices in , that is, .The sequence is the transmission degree sequence of , and is the diagonal matrix of vertex transmissions of .
Note that(1)Transmission of a vertex is also called the distance degree or the first distance degree of .(2)If for , then is called a -transmission regular graph.

Definition 2. Let be a graph with vertex set, distance matrix , and transmission degree sequence such that . Then, the second transmission of vertex , denoted by , is defined to be , and is called the second transmission degree sequence of .

Definition 3. (see [3]). Let be a graph of order . The Wiener index of is defined asIn 1970, Gutman first proposed the concept of graph energy in [4]. The adjacency matrix of a graph is a matrix of order whose -entry is equal to unity if the vertices and are adjacent and is equal to zero otherwise. Since is real and symmetric, all eigenvalues of are real, denoted by , also known as the eigenvalues of . The energy of graph is . Let , where for . The matrices and are called the Laplacian matrix and the signless Laplacian matrix of graph , respectively. For more research on Laplacian matrix and signless Laplacian matrix, refer to [58]. Aouchiche and Hansen [9, 10] introduced the distance Laplacian matrix and the distance signless Laplacian matrix of graph .
In 2019, Cui et al. [11] proposed the generalized distance matrix by using the convex linear combination of and , where . As you can see, , , , and . Since the matrix is real and symmetric, all its eigenvalues are real, denoted by , which are called the generalized distance eigenvalues of , and the generalized distance spectral radius of is defined as .

Definition 4. (see [12]). Let be a graph of order . The generalized distance energy of can be thought of as the mean deviation of the values of the generalized distance eigenvalues of , namely, .
The study of generalized distance spectrum was proposed by Cui et al. in [11]. They established some basic spectral properties of the generalized distance matrix of graphs, obtained the bounds of the generalized distance spectral radius, and determined the graphs with the minimum generalized distance spectral radius in all connected bipartite graphs with fixed vertices. In [13], the authors obtained some upper and lower bounds for the second largest eigenvalue of the generalized distance matrix of graphs, in terms of various graph parameters, and the graphs attaining the corresponding bounds are characterized. In [14], the authors obtained an upper bound for the smallest generalized distance eigenvalue in terms of different graph parameters. In particular, they showed that this upper bound is better than the upper bound obtained by Cui et al. In [14], the authors established the relations between the smallest eigenvalues of , , and and obtained sharp bounds for the smallest eigenvalue of in terms of various graph parameters. In [12, 1517], Alhevaz et al. established some new sharp bounds for the generalized distance spectral radius of by using different graph parameters and characterized the extremal graphs. Then, they obtained new bounds for the -th generalized distance eigenvalue. Moreover, through the eigenvalues of adjacency matrix and some auxiliary matrices, they studied the generalized distance spectrum of graphs obtained by generalization of the join graph operation. In 2020, they defined the generalized distance energy in [12] and gave the upper and lower bounds of the generalized distance energy. In [18, 19], Pirzada et al. obtained the bounds of the generalized distance spectral radius of bipartite graphs by using some parameters of graphs and characterized their extremal graphs. It was proved that for , the complete bipartite graph has the minimum generalized distance energy among all connected bipartite graphs, and for , the star has the minimum generalized distance energy among all trees. In addition, the generalized distance spectrum, generalized distance energy and the spectral spread of generalized distance matrix are also studied. For some recent results on spread of generalized distance matrix, we refer the readers to [20, 21] and the references therein. Inspired by the above literature, in this paper, we further study the generalized distance spectral radius and generalized distance energy.
The rest of the paper is organized as follows. In Section 2, several lemmas are given. In Section 3, the new lower and upper bounds of the generalized distance spectral radius are obtained according to the distance between the vertices and some parameters of the graph. In Section 4, we obtain new bounds on the generalized distance energy in terms of spectral radius and parameters that depend on the distance between the vertices and the order of the graph.

2. Lemmas

In this section, we give some definitions and lemmas to prepare for subsequent proofs.

Lemma 1 (see [22]). If is a nonnegative real matrix of order , then its spectral radius is an eigenvalue of and it has an associated nonnegative eigenvector. Furthermore, if is irreducible, then is a simple eigenvalue of with an associated positive eigenvector.

Lemma 2 (Rayleigh–Ritz theorem [23]). If is a real symmetric matrix of order with eigenvalues , then for a nonzero vector ,with equality holding if and only if is an eigenvector of corresponding to .

Lemma 3 (Cauchy–Schwartz inequality). Let and be real numbers for all . Then,

Equality holds if and only if for all .

Lemma 4 (see [11]). Let be a graph with distance degree sequence . Then,

The equality holds if and only if is distance regular.

Lemma 5 (see [11]). Let be a simple connected graph, be the transmission of vertex , and be the second transmission of . Then,

The equality holds if and only if is a constant for all .

Lemma 6 (see [24]). Let the transmission degree sequence of graph be and the second transmission degree sequence of be . Then,

Each equality holds if and only if is a transmission regular graph.

3. Lower and Upper Bounds of Generalized Distance Spectral Radius

In this section, the matrix sequence is introduced according to the relationship between the transmission and the second transmission, and the bounds of the generalized distance spectral radius in Lemmas 5 and 6 are generalized by using the matrix sequence in Theorems 13.

Definition 5. (see [25]). For , the matrix sequence is defined as follows: fix , let , and for each , let .
Note that for , , and for .

Theorem 1. Let G be a connected graph of order , be a real number, and be an integer. Then,

The equality holds (for particular values of and ) if and only if is a constant for all .

Proof. Let be the unit positive Perron eigenvector of corresponding to . Let be the unit positive vector defined byNote thatWe obtainTherefore,Now we assume that the equality holds in (7). By (9), is a positive eigenvector corresponding to . From , we obtain that for .
Conversely, if , then for all . Hence,that is, is a eigenvalue of and is a eigenvector corresponding to . Because is a positive vector, applying Lemma 1, we obtain , andThis completes the proof.

Example 1. For , let be the graphs given in Figure 1. In particular, , , and are the star, path, and cycle on seven vertices, denoted by , , and , respectively.
We observe that is a transmission regular graph and is a transmission regular graph. In Table 1, we show the lower bounds for , using four decimal places.

Theorem 2. Let G be a connected graph of order and be an integer. Then,The equality holds if and only if , , and is a constant for all .

Proof. Let be the unit positive Perron eigenvector of corresponding to , whereLet , , and the -entry of matrix beTherefore,The and lines in (18) can be expressed byThen,that is,So,Therefore,For , for : Let . Then, . We know that and According to Lemma 1, is the largest eigenvalue of , and , so the equality holds in (15). On the contrary, if inequality (15) is equal, then can be obtained from (19) and (20), that is, . It means when and , is a constant for . This completes the proof.

Theorem 3. Let be a connected graph and be an integer. Then,

The equality holds if and only if, , and is a constant for all .

Proof. The proof method is the same as Theorem 2.

Example 2. We consider the graphs given in Example 1 and bounds for given in Lemma 6 and Theorems 1 and 2. Using four decimal places, we obtain the upper bounds for , as shown in Table 2.

Theorem 4. Let be a graph of order and size . If the diameter of is , thenwith equality holding if and only if is a transmission regular graph.

Proof. Let. Since , by Lemma 2, we havethat is, (26) holds.
If is a transmission regular graph, then , and so the equality in (26) holds. Conversely, if the quality in (26) holds, it is clear that is a transmission regular graph.

4. Upper Bound of Generalized Distance Energy

In this section, we obtain new bounds on the generalized distance energy in terms of spectral radius and parameters that depend on the distance between the vertices and the order of the graph. Let be a graph of order with generalized distance eigenvalues . Let for . Then, are called the auxiliary eigenvalues of [19]. It is easy to see that , and

Denote , , and .

Theorem 5. Let be a connected graph of order . Then,

Proof. Let be the generalized distance eigenvalues of , and for . By Lemma 3, for , we getsoWe now consider the function with . For , ,and is monotonically decreasing. For , ,and is monotonically increasing. Thus, we have the following results.(i)If, then (ii)If, then (iii)If, then This completes the proof.

Lemma 7. If is a graph of diameter , then

Proof. Since there are elements equal to and elements equal to in , we haveThe lemma holds.

Theorem 6. If is a -regular graph of diameter , then

Proof. Since is a -regular graph of diameter 2, by Theorem 4, , and Applying Lemma 3 to and , we getBy Lemma 7,that is,Since and , (37) givesThe theorem follows.

5. Conclusions

In this paper, some new lower and upper bounds of the generalized distance spectral radius are obtained in terms of the distance between the vertices and some parameters of the graph. Meanwhile, we obtain new bounds on the generalized distance energy in terms of spectral radius and parameters that depend on the distance between the vertices and the order of the graph.

Data Availability

All data, models, and codes generated or used during the study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by the Shanxi Scholarship Council of China (no. 201901D211227).