New Bounds for the Generalized Distance Spectral Radius/ Energy of Graphs

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Introduction
Troughout this paper, we consider simple, connected, and fnite graphs. Let G be such graph with vertex set V(G) � v 1 , v 2 , . . . , v n and edge set E(G). Let d v i denote the degree of vertex v i and N(v i ) denote the neighbor set of v i . Te distance between vertices v i and v j in G is the length of the shortest path connecting v i to v j , which is denoted as d(v i , v j ). Te distance matrix [1,2] of G is an n × n matrix Defnition 1. Let G be a graph with vertex set V(G) � v 1 , v 2 , . . . , v n . Te transmission of vertex v i , denoted by Tr G (v i ) or Tr i , is defned to be the sum of the distances from v i to all vertices in G, that is, Tr G (v i ) � Tr i � u∈V(G) d(u, v i ).Te sequence Tr 1 , Tr 2 , . . . , Tr n is the transmission degree sequence of G, and Tr(G) � diag(Tr 1 , Tr 2 , . . . , Tr n ) is the diagonal matrix of vertex transmissions of G.
Note that (1) Transmission of a vertex v is also called the distance degree or the frst distance degree of v.
Defnition 2. Let G be a graph with vertex setV(G) � v 1 , v 2 , . . . , v n , distance matrix D(G), and transmission degree sequence Tr 1 , Tr 2 , . . . , Tr n such that Tr 1 ≥ Tr 2 ≥ . . . ≥ Tr n . Ten, the second transmission of vertex v i , denoted by T i , is defned to be T i � n j�1 d ij Tr j , and T 1 , T 2 , . . . , T n is called the second transmission degree sequence of G.
Defnition 3 (see [3]). Let G be a graph of order n. Te Wiener index of G is defned as In 1970, Gutman frst proposed the concept of graph energy in [4]. Te adjacency matrix A(G) of a graph G is a matrix of order n whose (i, j)-entry is equal to unity if the vertices v i and v j are adjacent and is equal to zero otherwise. Since A(G) is real and symmetric, all eigenvalues of A(G) are real, denoted by μ 1 , μ 2 , . . . , μ n , also known as the ei- where are called the Laplacian matrix and the signless Laplacian matrix of graph G, respectively. For more research on Laplacian matrix and signless Laplacian matrix, refer to [5][6][7][8]. Aouchiche and Hansen [9,10] introduced the distance Laplacian matrix D L (G) � Tr(G) − D(G) and the distance signless Laplacian matrix D Q (G) � Tr(G) + D(G) of graph G.
In 2019, Cui et al. [11] proposed the generalized distance matrix D α (G) � αTr(G) + (1 − α)D(G) by using the convex linear combination of Tr(G) and D(G), where 0 ≤ α ≤ 1. As you can see, Since the matrix D α (G) is real and symmetric, all its eigenvalues are real, denoted by λ 1 , λ 2 , . . . , λ n , which are called the generalized distance eigenvalues of G, and the generalized distance spectral radius of G is defned as ρ(D α (G)) � max 1≤i≤n λ i .
Defnition 4 (see [12]). Let G be a graph of order n. Te generalized distance energy of G can be thought of as the mean deviation of the values of the generalized distance eigenvalues of G, namely, E D α (G) � n i�1 |λ i − 2αW(G)/2n|. Te study of generalized distance spectrum was proposed by Cui et al. in [11]. Tey 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 fxed 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 λ n in terms of diferent 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 D α (G), D(G), and D Q (G) and obtained sharp bounds for the smallest eigenvalue λ n of D α (G) in terms of various graph parameters. In [12,[15][16][17], Alhevaz et al. established some new sharp bounds for the generalized distance spectral radius of G by using diferent graph parameters and characterized the extremal graphs. Ten, they obtained new bounds for the k-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 defned 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 α ∈ (1/2, 1), the complete bipartite graph has the minimum generalized distance energy among all connected bipartite graphs, and for α ∈ (0, 2n/3n − 2), 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.
Te 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.

Lemmas
In this section, we give some defnitions and lemmas to prepare for subsequent proofs.
Lemma 2 (Rayleigh-Ritz theorem [23]). If B is a real symmetric matrix of order n with eigenvalues λ 1 ≥ λ 2 ≥ . . . ≥ λ n , then for a nonzero vector X, with equality holding if and only if X is an eigenvector of B corresponding to λ 1 .

Lemma 3 (Cauchy-Schwartz inequality).
Let a k and b k be real numbers for all 1 ≤ k ≤ n. Ten, Equality holds if and only if a j b k � a k b j for all 1 ≤ k, j ≤ n .
Lemma 4 (see [11]). Let G be a graph with distance degree sequence Tr 1 , Tr 2 , . . . , Tr n . Ten, Te equality holds if and only if G is distance regular.
Lemma 5 (see [11]). Let G be a simple connected graph, Tr i be the transmission of vertex v i , and T i be the second transmission of v i . Ten, Te equality holds if and only if αTr i + (1 − α)T i /Tr i is a constant for all i � 1, 2, . . . , n.
Lemma 6 (see [24]). Let the transmission degree sequence of graph G be Tr 1 , Tr 2 , . . . , Tr n and the second transmission degree sequence of G be T 1 , T 2 , . . . , T n . Ten, Each equality holds if and only if G is a transmission regular graph.

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 Teorems 1-3.
Defnition 5 (see [25]). For i � 1, 2, . . . , n, the matrix sequence Theorem 1. Let G be a connected graph of order n, β be a real number, and t be an integer. Ten, Te equality holds (for particular values of β and t) if and Proof. Let X � (x 1 , x 2 , . . . , x n ) T be the unit positive Perron eigenvector of D α (G) corresponding to ρ(D α (G)). Let Y be the unit positive vector defned by Note that Mathematical Problems in Engineering 3 We obtain Terefore, Now we assume that the equality holds in (7). By (9), Y is a positive eigenvector corresponding to ρ(D α (G)). From that is, k is a eigenvalue of D α (G) and Y is a eigenvector corresponding to k. Because Y is a positive vector, applying Lemma 1, we obtain k � ρ(D α (G)), and Tis completes the proof. □ Example 1. For i � 1, . . . , 6, let G i be the graphs given in Figure 1. In particular, G 4 , G 5 , and G 6 are the star, path, and cycle on seven vertices, denoted by S 7 , P 7 , and C 7 , respectively.
We observe that G 1 is a 7− transmission regular graph and G 6 is a 12− transmission regular graph. In Table 1, we show the lower bounds for ρ(D α (G)), using four decimal places.

Theorem 2.
Let G be a connected graph of order n and t be an integer. Ten, Te equality holds if and only if t � 1, β � 1, and M (1) i is a constant for all i � 1, 2, . . . , n.
Proof. Te proof method is the same as Teorem 2. □ Example 2. We consider the graphs G 1 , G 2 , G 3 , G 4 , G 5 , G 6 given in Example 1 and bounds for ρ(D α (G)) given in Lemma 6 and Teorems 1 and 2. Using four decimal places, we obtain the upper bounds for ρ(D α (G)), as shown in Table 2.
with equality holding if and only if G is a transmission regular graph.

Mathematical Problems in Engineering
Theorem 5. Let G be a connected graph of order n. Ten, Proof. Let λ 1 ≥ λ 2 ≥ . . . ≥ λ n be the generalized distance eigenvalues of G, and θ i � λ i − 2αW(G)/n for i � 1, 2, . . . , n. By Lemma 3, for i � 1, 2, . . . , n, we get so We now consider the function is monotonically increasing. Tus, we have the following results.

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
Te authors declare that they have no conficts of interest.