Sharp Bounds on (Generalized) Distance Energy of Graphs

Given a simple connected graph G, let D ( G ) be the distance matrix, D L ( G ) be the distance Laplacian matrix, D Q ( G ) be the distance signless Laplacian matrix, and T r ( G ) be the vertex transmission diagonal matrix of G. We introduce the generalized distance matrix D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where α ∈ [ 0 , 1 ] . Noting that D 0 ( G ) = D ( G ) , 2 D 1 2 ( G ) = D Q ( G ) , D 1 ( G ) = T r ( G ) and D α ( G ) − D β ( G ) = ( α − β ) D L ( G ) , we reveal that a generalized distance matrix ideally bridges the spectral theories of the three constituent matrices. In this paper, we obtain some sharp upper and lower bounds for the generalized distance energy of a graph G involving different graph invariants. As an application of our results, we will be able to improve some of the recently given bounds in the literature for distance energy and distance signless Laplacian energy of graphs. The extremal graphs of the corresponding bounds are also characterized.


Introduction
We study in this paper simple connected graphs G = (V, E) with V(G) = {v 1 , v 2 , . . . , v n } being the vertex set and E(G) being the edge set. The order of G is denoted by |V(G)| = n and the size of G is denoted by |E(G)| = m. Let N(v) be the neighborhood of a vertex v in V(G). Let G represent the complement of G. Some classical graphs such as the complete graph, complete bipartite graph, path, and cycle are denoted by K n , K s,t , P n , and C n , respectively. The degree of v is denoted by d G (v) or simply d v . The adjacency matrix is A(G) = (a ij ) with Deg(G) = diag(d 1 , d 2 , . . . , d n ) being the diagonal degree matrix with d i = d G (v i ), i = 1, 2, . . . , n. The Laplacian and signless Laplacian matrices are signified by L(G) = Deg(G) − A(G) and Q(G) = Deg(G) + A(G), respectively. Their spectra are arranged as 0 = µ n ≤ µ n−1 ≤ · · · ≤ µ 1 and 0 ≤ q n ≤ q n−1 ≤ · · · ≤ q 1 , respectively.
The energy of a graph [21] as a mathematical chemistry concept was put forward by Ivan Gutman. In chemistry, the energy is used to approximate the total Π-electron energy of a molecule. Let λ 1 , λ 2 , . . . , λ n be the adjacency eigenvalues of a graph G. The energy of a graph G, denoted by E(G), [22] for an updated survey). Recently, other kinds of energies of a graph have been defined and studied. We recall some of them. Let ρ D 1 ≥ ρ D 2 ≥ · · · ≥ ρ D n and ρ L 1 ≥ ρ L 2 ≥ · · · ≥ ρ L n and also ρ Q 1 ≥ ρ Q 2 ≥ · · · ≥ ρ Q n represent the distance, distance Laplacian, and distance signless Laplacian eigenvalues, respectively. The distance energy of a graph G was introduced in [23] as For some recent results on the distance energy of a graph, we refer to [10] and the references therein. In addition, the concept of distance Laplacian and distance signless Laplacian energies were introduced in [7,10,24], respectively, as follows. The distance Laplacian energy of a graph G is defined by taking into consideration of distance Laplacian spectrum deviations as Similarly, the distance signless Laplacian energy of a graph G is defined as follows: For some recent papers on E L (G) and E Q (G), we refer to [7,10,25], and for other recent papers regarding the energy of a matrix with respect to different graph matrices; see [11,12,23,[26][27][28][29] and the references therein. Motivated by the definitions of E L (G) and E Q (G), Alhevaz et al. [17] recently defined the generalized distance energy of G as the average deviation of generalized distance spectrum: Tr 2 i , hence by the definition of Θ i , one can easily see that From the above definition, . Thus, exploring the properties of E D α (G) and its dependency with parameter α could give us a unified picture of the spectral properties of distance (signless Laplacian) energy of graphs.
The rest of the paper is structured as follows. In Section 2, for α ∈ [0, 1], we obtain some sharp lower bounds for the generalized distance energy E D α (G) of a connected graph G resorting to Wiener index W(G), transmission degrees, and the parameter α ∈ [0, 1]. The graphs attaining the corresponding bounds are also characterized. In Section 3, we obtain sharp upper bounds for the generalized distance energy E D α (G) involving diameter d, minimum degree δ, Wiener index W(G), as well as transmission degrees. Some extremal graphs that attain these bounds are determined in this section. As an application of our results in Section 3, we will be able to improve some recently given upper bounds for distance (signless Laplacian) energy in [25].

Lower Bounds for E D α (G)
In this section, we give some sharp lower bounds for E D α (G) in terms of different graph parameters. Firstly, we include some previous known results that will play a pivotal role in the rest of the paper. Lemma 1 ([16]). If G is a connected graph, then where the equality holds if and only if G is transmission regular.

Lemma 2.
Recall that {Tr 1 , Tr 2 , . . . , Tr n } constitutes the transmission degrees. We have where the equality holds if and only if G is transmission regular.
Moreover, if 1 2 ≤ α ≤ 1, the equality holds if and only if G is transmission regular.

Remark 1.
Keeping all of the notations from Lemma 3, we have In fact, as we always have ∑ n i=1 T i = ∑ n i=1 Tr 2 i , and also applying the Cauchy-Schwarz inequality, we have

Lemma 4 ([30]
). Assume that a i and b i , i = 1, 2, . . . , n, are positive real numbers. We have where M 1 = max 1≤i≤n a i , M 2 = max 1≤i≤n b i , m 1 = min 1≤i≤n a i and m 2 = min 1≤i≤n b i .

Lemma 5 ([31]
). If b 1 , b 2 , . . . , b n are positive numbers, then: for any real numbers a 1 , a 2 , . . . , a n . Equality holds if and only if a i b i are equal for all i.
The following lemma characterizes the graphs with exactly two distinct generalized distance eigenvalues.

Lemma 6.
A connected graph G possesses precisely two different D α (G) eigenvalues if and only if it is a complete graph.
Proof. The proof is analogous to that of (Lemma 2.10 [32]).
Our first lower bound for the generalized distance energy E D α (G) relies on the Wiener index W(G) as well as the transmission degrees.

Theorem 1.
Assume that G is a connected graph with n > 1 nodes. We have where the equality holds if and only if G is a complete graph.
Proof. By the Cauchy-Schwarz inequality, we have that is, Since Suppose that equality holds. Then, from equality in (3), we get Hence, G has exactly one distinct D α -eigenvalue or G has exactly two distinct D α -eigenvalues. In view of Lemma 6, we get G ∼ = K n , and the proof is complete.
Next, we give a lower bound for E D α (G) utilizing only the Wiener index W(G).

Theorem 2.
Assume that G is a connected graph having n vertices. Suppose that α ≤ 1 − n 2W(G) . Then, where The equality in (4) holds if and only if α = 0 and G ∼ = K n or G is a k-transmission regular graph with three different generalized distance eigenvalues represented as k, αk + 1 and αk − 1.

Proof.
We construct a function n > 0, with equality holding if and only if x − 2αW(G) n = 1. With these at hand, we get From Lemma 1, we know that ∂ 1 ≥ 2W(G) n . Consider the function In the light of these results and (5), we derive (4). Suppose the equality holds in (4). Then, ∂ 1 = 2W(G) n and so, by Lemma 1, G is a transmission regular graph. From equality in (5), we get ∂ i − 2αW(G) n = 1, for i = 2, 3, . . . , n. This gives that can have no more than two different values and we obtain the following: n for i = 2, 3, . . . , n, yielding that G has a pair of different generalized distance eigenvalues, Thus, by Lemma 6, G is complete. As the generalized distance eigenvalues of K n are spec Thus, by Lemma 6 , G is complete, which is true for α = 0, giving that equality occurs in this case for α = 0 and if and only if G ∼ = K n .
(iii) In this case, let, for some t, On the other hand, suppose that G ∼ = K n . Noting the generalized distance eigenvalues of K n are spec(K n ) = {n − 1, αn − 1 [n−1] }, and 2αW(K n ) n = α(n − 1), we obtain that the equality holds in (4). In addition, if G is k-transmission regular graph possessing three different generalized distance eigenvalues k, αk + 1 and αk − 1, then the equality is true. Now, by Remark 1 and proceeding similarly to Theorem 2, we obtain the following lower bound for E D α (G) using the transmission degrees as well as the second transmission degrees. Theorem 3. Let G be a connected graph with n vertices and α ≤ 1 − n 2W(G) . Then, where The equality in (7) holds if and only if α = 0 and G ∼ = K n or G is a k-transmission regular graph with three different generalized distance eigenvalues, namely k, αk + 1 and αk − 1.
We conclude this section by giving another sharp lower bound on the generalized distance energy.

Theorem 4. Let G be connected with n vertices and ρ
Equality holds if and only if either G is a complete graph or a graph with exactly three distinct D α -eigenvalues.
Proof. Applying the Cauchy-Schwarz inequality, we obtain that is, Since Let us consider a function Then, In order to calculate the extreme point, we require f (x) = 0. This implies At this point, Therefore, the function f (x) reaches a minimum at x = det D α (G) − 2αW(G) n 1 n , and the minimum value is However, Suppose that β is the integer such that ∂ β ≥ 2αW(G) n and ∂ β+1 ≤ 2αW(G) n . By Lemma 5, we have Then, for ρ n ≥ (2α − 1) 2W(G) n , and α = 1, we have Therefore, the function f (x) is increasing in the interval and then Hence, The first half of the proof is complete. Now, suppose equality holds in (8). In this situation, From equality in (9), we get Therefore, Hence, ∂ i − 2αW(G) n can have at most two distinct values and we arrive at the following: (i) G has only one D α -eigenvalue. Then, G ∼ = K 1 .
(iii) G possesses precisely three different D α -eigenvalues. Therefore, Then, we get that G is a graph with exactly three distinct D α -eigenvalues, and the result follows.
Some well-known special graphs include Hamming graph H(n, d), the complete split graph CS t,n−t and the lexicographic product graph G[H]. For H(n, d), its vertex set is represented by X n with d elements in X. If precisely one coordinate of two vertices are different, then they are adjacent. In particular, H(n, 2) becomes the cube Q n . The graph CS t,n−t is composed of a clique over t vertices and an independent set of n − t vertices. The vertices in cliques are required to be neighbors of each vertex in the independent set. G[H] has the vertex set V(G) × V(H) and two vertices are adjacent whenever their first coordinates are adjacent in G or they have the same first coordinate, but their second coordinates are adjacent in H.

Remark 2.
Note that there are some graphs that have exactly three or four distinct generalized distance eigenvalues. For example, the star graph, the cycle C 4 , the cycle C 5 , and square of the hypercube of dimension n, Q 2 n have exactly three distinct generalized distance eigenvalues. In addition, the complete bipartite graph K a,b , where a, b ≥ 3, a + b = n, the complete split graph CS t,n−t , the complement of an edge K n − e and the closed fence C 4 [K 2 ] have four different generalized distance eigenvalues.
Although we have given in Remark 2 some special classes of graphs with exactly three and exactly four distinct generalized distance eigenvalues, we were unable to giving a complete characterization of such graphs. It will be an interesting problem to characterize all the connected graphs having precisely three or four distinct generalized distance eigenvalues. Therefore, we leave the following problems: Problem 1. Characterize all the connected graphs having precisely three different generalized distance eigenvalues.

Problem 2.
Characterize all the connected graphs having precisely four different generalized distance eigenvalues.

Upper Bounds for E D α (G)
In this section, we obtain some sharp upper bounds for the generalized distance energy E D α (G) of a connected graph G by using diameter d, minimum degree δ, Wiener's index W(G), as well as transmission degrees. The extremal graphs are characterized accordingly. As an application of our results, we will be able to improve some recently given upper bounds for distance energy and distance signless Laplacian energy of a graph G in [25].

Remark 3. Following [25], we have
Tr 2 i , then we get Theorem 5. Let G be a connected graph of order n. If 0 ≤ α ≤ 1 2 , then Equality holds if and only if either G is a complete graph or G is a graph with exactly three distinct D α -eigenvalues.

Proof. Applying the Cauchy-Schwarz inequality, we have
Hence, Thus, We construct a function It follows from straightforward calculations that the function f (x) monotonically decreases for x ≥ P n . Now, by Lemma 2, Remark 3, and inequality we have and hence The first half of the proof is complete.
If the equality holds in (10), we see that From equality in (11), we get then we have Hence, ∂ i − 2αW(G) n can have no more than a pair of different values and we arrive at the following: (i) G has only one D α -eigenvalue. Then, G ∼ = K 1 .
(ii) G has precisely a pair of different D α -eigenvalues. Thanks to Lemma 6, G ∼ = K n .
(iii) G has precisely three different D α -eigenvalues. We have Then, we obtain that G is a graph with three distinct D α -eigenvalues.
The following result gives an upper bound for the generalized distance energy E D α (G) using Wiener's index W(G), diameter d as well as minimum degree δ. Corollary 1. Let G be connected having n vertices. If 0 ≤ α ≤ 1 2 , then , where the equality holds if and only if either G is a complete graph or G is a graph with precisely three different D α -eigenvalues.
Hence, if 0 ≤ α ≤ 1 2 , then, by Theorem 5, we get Hence, from the upper bound of Theorem 5, the first part of the proof is done. The rest of the proof follows Theorem 5.
Since for any i, we have n − 1 ≤ Tr i ≤ n(n−1) 2 , hence one can analogously show the following theorem.

Corollary 2.
Let G be connected possessing n vertices. If 0 ≤ α ≤ 1 2 , then The equality holds if and only if either G is a complete graph or G is a graph with exactly three distinct D α -eigenvalues.

Remark 4.
If G is connected possessing positive generalized distance eigenvalues, then for 0 ≤ α ≤ 1 2 , we have where a 1 , . . . , a n are positive real numbers (see [25]); hence, we get Hence, it can be easily seen that the inequality occurs in (13). On the other hand, since ∑ n i=1 Tr i 2 ≤ n ∑ n i=1 Tr 2 i , we obtain Theorem 6. Let G be connected having n ≥ 3 vertices.
(i) If α = 0, then The equality holds if and only if G possesses precisely three or four different D α -eigenvalues.
Taking derivatives on f (x, y) with respect to x and y, we have In order to calculate the extreme values, we set f x = 0 and f y = 0, This yields x = y = P n . At this point, the values of f xx , f yy , f xy and ∆ = f xx f yy − f 2 xy are Hence, f (x, y) has maximum value at this point, and accordingly f P n , P n = √ nP.
Nevertheless, f (x, y) decreases in the intervals We examine the following two situations: (see [25]), we obtain In addition, we obtain Hence, Then, Therefore, Again by Remark 4 and as Then, Therefore, The rest of the proof follows from Theorem 4.

Remark 5.
Keeping all of the notations from Theorem 6, and taking then it is clear that f (x, y) ≤ h(x, y) for all (x, y) in the given region of x and y. For 0 ≤ α ≤ 1 2 , along . The function f 2(1−α)W(G) n , y decreases in . By Remark 4, we have Thus, Hence, The following upper bound was proved in in [25]: Remark 6. For α = 0, it is easily seen by Remark 5 that the upper bound in Theorem 6 improves that presented in (14).
In addition, the following upper bound for the distance signless Laplacian energy E Q (G) was obtained in [25]: Remark 7. For α = 1 2 , it is not difficult to see by Remark 5 that the upper bound shown in Theorem 6 improves that presented in (15).
We recall the following lemma. Lemma 7 (Theorem 2.11 [8]). Let G have n > 1 vertices. For the largest and second largest generalized distance eigenvalues ∂ 1 and ∂ 2 of G, we have Equality holds if and only if G is a graph with exactly three or exactly four distinct D α -eigenvalues.
We conclude with the following upper bound by using only the Wiener index W(G).
Theorem 7. Let G be connected having n > 1 vertices. If ∂ 2 ≥ 2αW(G) n , then The equality holds if and only if G is a graph with precisely three or four different D α -eigenvalues.
The rest of the proof follows similarly as Theorem 4.

Conclusions
The notion of generalized distance energy of a graph G was first motivated in Alhevaz et al. [17] as the average deviation of generalized distance spectrum: where W(G) is Wiener's index. Arguably, the distance and the distance signless Laplacian play a pivotal role in mathematics as they offer more information than the classical binary adjacency matrix.
In this work, we along this line further investigate the energy of a generalized distance matrix. It forms a natural extension of the theory of distance energy as well as distance signless Laplacian energy. The spectral properties of these relevant individual combinatorial matrices can be derived as special situations in the framework of a generalized distance matrix. We developed some properties of E D α (G) by establishing new inequalities including sharp upper and lower bounds linking a range of invariants such as diameter, extreme degree, Wiener's index as well as transmission degrees. Existing bounds in the literature have been improved and extremal graphs have been determined. For future work, it would be desirable to derive some other sharp bounds for the generalized distance energy leveraging a variety of graph invariants.