ISSN 2148-838X

J. Algebra Comb. Discrete Appl.
10(2) • 73–86

Received: 26 November 2020
Accepted: 31 December 2022

Journal of Algebra Combinatorics Discrete Structures and Applications

Some upper and lower bounds for Dα-energy of graphs

Research Article

Abdollah Alhevaz, Maryam Baghipur, Ebrahim Hashemi, Yilun Shang

Abstract: The generalized distance matrix of a connected graph G, denoted by Dα(G), is defined as Dα(G) =
αTr(G) + (1 − α)D(G), 0 ≤ α ≤ 1. Here, D(G) is the distance matrix and Tr(G) represents
the vertex transmissions. Let ∂1 ≥ ∂2 ≥ · · · ≥ ∂n be the eigenvalues of Dα(G) and let W(G) be the

Wiener index. The generalized distance energy of G can be defined as EDα(G) =
n∑
i=1

∣∣∣∣∂i − 2αW(G)n
∣∣∣∣.

In this paper, we develop some new theory regarding the generalized distance energy EDα(G) for a
connected graph G. We obtain some sharp upper and lower bounds for EDα(G) connecting a wide
range of parameters in graph theory including the maximum degree ∆, the Wiener index W(G),
the diameter d, the transmission degrees, and the generalized distance spectral spread DαS(G). We
characterized the special graph classes that attain the bounds.

2010 MSC: 05C50, 05C12, 15A18

Keywords: Generalized distance matrix, Generalized distance energy, Distance (signless Laplacian) matrix, Transmission
regular graph, Generalized distance spectral spread

1. Introduction

We consider simple connected graphs in this paper. Let G = (V (G),E(G)) be a graph with V (G) =
{v1,v2, . . . ,vn} being its vertex set and E(G) its edge set. n = |V (G)| is called the order and m = |E(G)|
the size. The neighborhood of a vertex v is the collection of vertices adjacent to it and is denoted by
N(v). dG(v) or simply dv is the degree of v, meaning the cardinality of its neighborhood. A regular graph
has all degrees the same. The adjacency matrix A(G) = A = (aij) of G has (i,j)-element one if vi is

Abdollah Alhevaz; Faculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box: 316-
3619995161, Shahrood, Iran(email: a.alhevaz@shahroodut.ac.ir).
Maryam Baghipur; Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran (email:

maryamb8989@gmail.com).
Ebrahim Hashemi; Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran (email:

eb_hashemi@shahroodut.ac.ir).
Yilun Shang (Corresponding Author); Department of Computer and Information Sciences, Northumbria Uni-

versity, Newcastle NE1 8ST, UK (email: yilun.shang@northumbria.ac.uk).

73

https://orcid.org/0000-0001-6167-607X
https://orcid.org/0000-0002-9069-9243
https://orcid.org/0000-0002-8673-9556
https://orcid.org/0000-0002-2817-3400


A. Alhevaz et.al. / J. Algebra Comb. Discrete Appl. 10(2) (2023) 73–86

adjacent to vj and zero if not. Hence, A is an n by n symmetric matrix. The degree diagonal matrix
is Deg(G) = diag(d1,d2, . . . ,dn). Two well known matrices associated with A are the Laplacian L(G) =
Deg(G) − A(G) and the signless Laplacian Q(G) = Deg(G) + A(G). They are symmetric and positive
semi-definite. Their spectra are organized as 0 = µn ≤ µn−1 ≤ ··· ≤ µ1 and 0 ≤ qn ≤ qn−1 ≤ ··· ≤ q1,
respectively.

The length of a shortest path between two vertices u and v is commonly known as the distance and
is denoted by duv. If we take maximum among all such distances in a graph, we have the diameter.
The matrix D(G) = (duv)u,v∈V (G) is called the distance matrix of G. For a comprehensive survey of
recent results on distance matrix and its spectrum, we refer the reader to [6]. The sum of the distances
from v to all other vertices in G is called the transmission and it is denoted by TrG(v) =

∑
u∈V (G)

duv. If

TrG(v) = k for all v, then G is called k-transmission regular. The well known Wiener index is defined
by W(G) = 1

2

∑
v∈V (G)

TrG(v) and is also called transmission of a graph. The transmission TrG(vi) or

shortly Tri for a vertex vi is also called the transmission degree. The transmission degree sequence of G
is {Tr1,Tr2, . . . ,Trn}.

The distance Laplacian and the distance signless Laplacian matrices of connected graphs have been
introduced in M. Aouchiche and P. Hansen [7]. Two matrices are of utmost relevance: the distance Lapla-
cian matrix DL(G) = Tr(G)−D(G) and the distance signless Laplacian matrix DQ(G) = Tr(G)+D(G).
Here, Tr(G) = diag(Tr1,Tr2, . . . ,Trn) characterizes the vertex transmission of G. Many important spec-
tral properties of these matrices have been intensively explored in the recent years; see e.g. [4, 5, 7, 8].

An effort has been made in [9] to merge the different spectra theory of distance matrix, distance
Laplacian etc. by introducing the so-called generalized distance matrix Dα(G), where Dα(G) = αTr(G)+
(1 − α)D(G), for 0 ≤ α ≤ 1. Note that D0(G) = D(G), 2D1

2
(G) = DQ(G), D1(G) = Tr(G) and

Dα(G)−Dβ(G) = (α−β)DL(G). Therefore, the spectral properties of individual graph matrices can be
reproduced from the spectral theory of generalized distance matrix. We will re-arrange the eigenvalues of
Dα(G) as ∂1 ≥ ∂2 ≥ ···≥ ∂n. The largest eigenvalue ∂1 is referred to as the generalized distance spectral
radius. When no confusion will be caused, we simply write ∂(G). The spectral properties of Dα(G) have
attracted much more attention of the researchers. For some recent works we refer to [1–3, 9, 11, 22, 23]
and the references therein.

The topic of graph energy [13] was put forward by Ivan Gutman. It is rooted in the theory of
mathematical chemistry. Assume the adjacency spectrum of a graph G is represented by λ1,λ2, . . . ,λn.

The graph energy is defined as E(G) =
n∑
i=1

|λi| [14]. Graph energy has been intensively studied in

mathematical chemistry and some early bounds for E(G) have been reported in e.g. [17]. Energy-like
graph invariants with respect to other matrices (in addition to the adjacency matrix) have been discussed
in [5, 10, 12, 15, 16, 21, 24].

This paper aims to study a new energy-like quantity on the basis of the eigenvalues of generalized
distance matrix Dα(G). We define auxiliary eigenvalues Θi corresponding to the generalized distance
eigenvalues as Θi = ∂i −

2αW(G)
n

. The following definition of generalized distance energy is given in [3],
which is inspired by the distance Laplacian energy EL(G) and the distance signless Laplacian EQ(G).
Namely,

EDα(G) =

n∑
i=1

∣∣∣∣∂i − 2αW(G)n
∣∣∣∣ = n∑

i=1

|Θi|,

which is the average deviation of the generalized distance eigenvalues.

Let t be the largest positive integer satisfying ∂t ≥
2αW(G)

n
. Let Mk(G) =

∑k
i=1 ∂i be the sum of k

74



A. Alhevaz et.al. / J. Algebra Comb. Discrete Appl. 10(2) (2023) 73–86

largest generalized distance eigenvalues. It is shown in [3] that

EDα(G) = 2

(
Mt −

2αtW(G)

n

)
= 2 max

1≤j≤n

(
j∑
i=1

∂i −
2αjW(G)

n

)
.

It can be seen that
∑n
i=1 Θi = 0. Noting

∑n
i=1 ∂i = 2αW(G) and

n∑
i=1

∂2i = trace[Dα(G)]
2 =

2(1 − α)2
n∑

1≤i<j≤n

(dij)
2 + α2

n∑
i=1

Tr2i , in view of Θi we obtain
∑n
i=1 Θ

2
i = 2ζ, where ζ = (1 −

α)2
∑n

1≤i<j≤n(dij)
2 + α

2

2

n∑
i=1

Tr2i −
2α2W2(G)

n
. Moreover, ED0 (G) = ED(G) and 2E

D 1
2 (G) = EQ(G).

The generalized distance graph energy can be viewed as merging the theories of distance graph energy and
distance signless Laplacian graph energy. It is therefore appealing to investigate EDα(G) and explore the
influence of the graph structure and parameter α on EDα(G). In [22], the authors defined the generalized
distance spectral spread as DαS(G) = ∂1 −∂n. Some lower and upper bounds for the parameter DαS(G)
can be found in [22].

In this paper, we continue on this line and study the generalized distance energy EDα(G) of a
connected graph G. We establish some sharp upper and lower bounds for EDα(G) using graph-theoretic
parameters including Wiener index W(G), maximum degree ∆, minimum degree δ, diameter d, as well
as generalized distance spectral spread DαS(G) and transmission degrees. We characterize the graphs
that attain our bounds.

2. New bounds on generalized distance energy of graphs

We start this section with some essential lemmas which will be used along the paper. The following
lemma characterizes the graphs with two generalized distance eigenvalues.

Lemma 2.1. A connected graph G has two different Dα(G) eigenvalues if and only if G is complete.

Proof. The proof is similar to the proof given in [15, Lemma 2], and is excluded.

Lemma 2.2. [9] Suppose that G is an n-vertex connected graph. Then

∂(G) ≥
2W(G)

n
,

with equality holding if and only if G is transmission regular.

Lemma 2.3. [18] Suppose that A is an n×n non-negative matrix with the spectral radius λ(A) and row
sums r1,r2, . . . ,rn. We have min1≤i≤n ri ≤ λ(A) ≤ max1≤i≤n ri. If A is irreducible, then one of the
equalities holds if and only if all the row sums of A are equal.

Since the i-th row sum of Dα(G) is ri = αTri + (1 −α)
∑n
j=1 dij = Tri, by Lemma 2.3 we have the

following statement.

Corollary 2.4. Suppose G is an n-vertex simple connected graph. Let Trmax and Trmin be the largest
and the smallest transmissions of G, respectively. We have Trmin ≤ ∂(G) ≤ Trmax. Furthermore, any
equality above holds if and only if G is transmission regular.

Our first aim is to giving some upper bounds for EDα(G). The following theorem presents an upper
bound for EDα(G). Some parameters like Wiener index W(G) and transmission degrees are used.

75



A. Alhevaz et.al. / J. Algebra Comb. Discrete Appl. 10(2) (2023) 73–86

Theorem 2.5. Suppose G is a graph of order n. We have

EDα(G) ≤

√√√√√n

2(1 −α)2 ∑

1≤i<j≤n

(dij)2 + α2
n∑
i=1

Tr2i −
4α2W2(G)

n


. (1)

The above equality holds if and only if G is a complete graph.

Proof. Let ∂1 ≥ ∂2 ≥ ··· ≥ ∂n be the eigenvalues of Dα(G). By applying the Cauchy-Schwarz
inequality to these n−1 vectors (1, 1, . . . , 1) and

(∣∣∣∂2 − 2αW(G)n ∣∣∣ ,∣∣∣∂3 − 2αW(G)n ∣∣∣ , . . . ,∣∣∣∂n − 2αW(G)n ∣∣∣) we
obtain, (

EDα(G) −
(
∂1 −

2αW(G)

n

))2
≤ (n− 1)

(
2(1 −α)2

∑
1≤i<j≤n

(dij)
2 + α2

n∑
i=1

Tr2i

−
4α2W2(G)

n
−
(
∂1 −

2αW(G)

n

)2 )
. (2)

Then

EDα(G) ≤ ∂1 −
2αW(G)

n

+

√√√√√(n− 1)

2(1 −α)2 ∑

1≤i<j≤n

(dij)2 + α2
n∑
i=1

Tr2i −
4α2W2(G)

n
−
(
∂1 −

2αW(G)

n

)2.
The function

f(x) = x +

√√√√√(n− 1)

2(1 −α)2 ∑

1≤i<j≤n

(dij)2 + α2
n∑
i=1

Tr2i −
4α2W2(G)

n
−x2




decreases if and only if x ≥

√
2(1−α)2

∑
1≤i<j≤n(dij)

2+α2
∑
n
i=1

Tr2
i
−4α

2W2(G)
n

n
. Therefore

EDα(G) ≤ f



√

2(1 −α)2
∑

1≤i<j≤n(dij)
2 + α2

∑n
i=1 Tr

2
i −

4α2W2(G)
n

n


 .

Hence the result follows.

Now, suppose equality holds in (1). Then

∂1 =

√
2(1 −α)2

∑
1≤i<j≤n(dij)

2 + α2
∑n
i=1 Tr

2
i −

4α2W2(G)
n

n
+

2αW(G)

n
.

From equality in (2), we get
∣∣∣∂2 − 2αW(G)n ∣∣∣ = ∣∣∣∂3 − 2αW(G)n ∣∣∣ = · · · = ∣∣∣∂n − 2αW(G)n ∣∣∣ and hence

n∑
i=2

(
∂i −

2αW(G)

n

)2
=

(
2(1 −α)2

∑
1≤i<j≤n

(dij)
2 + α2

n∑
i=1

Tr2i −
4α2W2(G)

n

−
(
∂1 −

2αW(G)

n

)2 )
,

76



A. Alhevaz et.al. / J. Algebra Comb. Discrete Appl. 10(2) (2023) 73–86

that is,

(n− 1)
(
∂i −

2αW(G)

n

)2
=

(
2(1 −α)2

∑
1≤i<j≤n

(dij)
2 + α2

n∑
i=1

Tr2i −
4α2W2(G)

n

−
(
∂1 −

2αW(G)

n

)2 )
.

Therefore for i = 2, . . . ,n, we get∣∣∣∣∂i − 2αW(G)n
∣∣∣∣ =

√
2(1−α)2

∑
1≤i<j≤n(dij)

2+α2
∑
n
i=1

Tr2
i
−4α

2W2(G)
n

−(∂1−
2αW(G)

n )
2

n−1

=

√
2(1−α)2

∑
1≤i<j≤n(dij)

2+α2
∑
n
i=1

Tr2
i
−4α

2W2(G)
n

n
.

Thus, we have ∂1 = P +
2αW(G)

n
and ∂i =

2αW(G)
n

+ P or ∂i =
2αW(G)

n
− P, where P =√

2(1−α)2
∑

1≤i<j≤n(dij)
2+α2

∑
n
i=1

Tr2
i
−4α

2W2(G)
n

n
. In the first case G has one distinct Dα-eigenvalue, which

is so if G = K1. In the second case G has two distinct eigenvalues namely, P +
2αW(G)

n
and 2αW(G)

n
−P

with multiplicity 1 and n−1, respectively and so by Lemma 2.1, we have G = Kn. Conversely, it is easy
to see that if G = Kn, the equality occurs. This completes the proof.

Corollary 2.6. Suppose G is a connected graph over n vertices. Assume the diameter is d. We have

EDα(G) ≤ n

√
(n− 1)

(
(1 −α)2d2 +

α2n2(n− 1)
4

−α2(n− 1)
)
. (3)

The above equality holds if and only if G ∼= K2.

Proof. Since dij ≤ d for i 6= j, and there are
n(n−1)

2
pairs of vertices in G, from the upper bound part

in Theorem 2.5, and also using the inequality Tri ≤
n(n−1)

2
, we conclude

EDα(G) ≤

√√√√√n

2(1 −α)2 ∑

1≤i<j≤n

(dij)2 + α2
n∑
i=1

Tr2i −
4α2W2(G)

n




≤

√
n

(
2(1 −α)2d2

n(n− 1)
2

+
α2n3(n− 1)2

4
−α2n(n− 1)2

)

= n

√
(n− 1)

(
(1 −α)2d2 +

α2n2(n− 1)
4

−α2(n− 1)
)
.

Equality occurs in the inequality Tri ≤
n(n−1)

2
if and only if G ∼= Pn. This together with Theorem 2.5,

gives that equality occurs in the inequality (3) if and only if G ∼= K2.

The following result provides an upper bound for EDα(G) via transmission degrees together with
Θ1 and Θn.

Theorem 2.7. Let G be a connected graph of order n. Then

EDα(G) ≤

√√√√√n

2(1 −α)2 ∑

1≤i<j≤n

(dij)2 + α2
n∑
i=1

Tr2i −
4α2W2(G)

n


− n

2

(
|Θ1|− |Θn|

)2
. (4)

77



A. Alhevaz et.al. / J. Algebra Comb. Discrete Appl. 10(2) (2023) 73–86

Proof. According to n
∑n
i=1 |Θi|

2 −
(∑n

i=1 |Θi|
)2

=
∑

1≤i<j≤n(|Θi|− |Θj|)
2, we derive that

n

n∑
i=1

|Θi|2 −

(
n∑
i=1

|Θi|

)2
≥
n−1∑
i=2

((
|Θ1|− |Θi|

)2
+
(
|Θi|− |Θn|

)2)
+
(
|Θ1|− |Θn|

)2
. (5)

Thanks to Jensen’s inequality (see [19]), we obtain

n−1∑
i=2

((
|Θ1|− |Θi|

)2
+
(
|Θi|− |Θn|

)2) ≥ n− 2
2

(
|Θ1|− |Θn|

)2
. (6)

By inequalities (5) and (6), we arrive at the inequality (4).

Remark 2.8. Since for every connected graph with the property |∂1| 6= |∂n|, we have√√√√√n

2(1 −α)2 ∑

1≤i<j≤n

(dij)2 + α2
n∑
i=1

Tr2i −
4α2W2(G)

n


− n

2
(|Θ1|− |Θn|)2

≤

√√√√√n

2(1 −α)2 ∑

1≤i<j≤n

(dij)2 + α2
n∑
i=1

Tr2i −
4α2W2(G)

n


,

it follows that the given upper bound in (4) is better than the given bound in (1).

Corollary 2.9. Let G be a connected graph satisfying |∂1| 6= |∂n|. Then

EDα(G) ≤
√
n

2


n

(
2(1 −α)2

∑
1≤i<j≤n(dij)

2 + α2
∑n
i=1 Tr

2
i −

4α2W2(G)
n

)
|Θ1|− |Θn|


 . (7)

Proof. Inequality (4) can be rewritten as

(EDα(G))2 +
n

2
(|Θ1|− |Θn|)2 ≤ n


2(1 −α)2 ∑

1≤i<j≤n

(dij)
2 + α2

n∑
i=1

Tr2i −
4α2W2(G)

n


 .

From the inequality between arithmetic and geometric means [19] we obtain

2

√
n

2
(EDα(G))2(|Θ1|− |Θn|)2 ≤ n


2(1 −α)2 ∑

1≤i<j≤n

(dij)
2 + α2

n∑
i=1

Tr2i −
4α2W2(G)

n


 ,

where from the inequality (7) follows.

Our next result presents a sharp upper bound on the generalized distance energy. Invariants like
transmission degrees and α are involved.

Theorem 2.10. Let G be an n-vertex connected graph with n ≥ 3. We have

EDα(G) ≤ (1 −α)
√∑n

i=1 Tr
2
i

n
+

(1 −α)
√∑n

i=1 Tr
2
i

n

+

√
(n− 2)

(
2ζ −

(1 −α)2
∑n
i=1 Tr

2
i

n
−

(1 −α)2
∑n
i=1 Tr

2
i

n2

)
. (8)

If equality holds in (8) then G must have exactly three or exactly four distinct Dα-eigenvalues.

78



A. Alhevaz et.al. / J. Algebra Comb. Discrete Appl. 10(2) (2023) 73–86

Proof. Using the Cauchy-Schwarz inequality on two (n− 2)-vectors (1, 1, . . . , 1︸ ︷︷ ︸
(n-2) times

) and

(∣∣∣∣∂2 − 2αW(G)n
∣∣∣∣ ,
∣∣∣∣∂3 − 2αW(G)n

∣∣∣∣ , . . . ,
∣∣∣∣∂n−1 − 2αW(G)n

∣∣∣∣
)
,

we have (
n−1∑
i=2

∣∣∣∣∂i − 2αW(G)n
∣∣∣∣
)2
≤

(
n−1∑
i=2

1

)(
n−1∑
i=2

∣∣∣∣∂i − 2αW(G)n
∣∣∣∣2
)
, (9)

and then

EDα(G) ≤
∣∣∣∣∂1 − 2αW(G)n

∣∣∣∣ +
∣∣∣∣∂n − 2αW(G)n

∣∣∣∣
+

√√√√(n− 2)
(

2ζ −
(
∂1 −

2αW(G)

n

)2
−
(
∂n −

2αW(G)

n

)2)
,

where 2ζ = 2(1 − α)2
∑

1≤i<j≤n(dij)
2 + α2

∑n
i=1 Tr

2
i −

4α2W2(G)
n

. Let x =
∣∣∣∂1 − 2αW(G)n ∣∣∣ and y =∣∣∣∂n − 2αW(G)n ∣∣∣. Define a function

f(x,y) = x + y +
√

(n− 2)(2ζ −x2 −y2).

Differentiating f(x,y) with respect to x and y, we have

fx = 1 −
x(n− 2)√

(n− 2) (2ζ −x2 −y2)
, fy = 1 −

y(n− 2)√
(n− 2) (2ζ −x2 −y2)

,

fxx = −
(
2ζ −y2

)√
n− 2√

(2ζ −x2 −y2)3
, fyy = −

(
2ζ −x2

)√
n− 2√

(2ζ −x2 −y2)3

and

fxy = −
xy
√
n− 2√

(2ζ −x2 −y2)3
.

For maxima or minima, fx = 0 and fy = 0 which implies x = y =
√

2ζ
n
. At this point the values of

fxx,fyy,fxy and ∆ = fxxfyy −f2xy are

fxx = −
(n− 1)

√
n− 2√

2(n−2)3ζ
n

≤ 0, fyy = −
(n− 1)

√
n− 2√

2(n−2)3ζ
n

≤ 0

fxy = −
√
n− 2√

2(n−2)3ζ
n

≤ 0, and ∆ =
n(n2 − 3n + 3)

2(n− 2)2ζ
≥ 0.

Therefore f(x,y) attains its maximum value at this point, hence f
(√

2ζ
n
,
√

2ζ
n

)
=
√

2nζ. However f(x,y)

decreases in the intervals
√

2ζ
n
≤ x ≤

√
ζ and 0 ≤ y ≤

√
2ζ
n
. Since 2

∑
1≤i<j≤n(dij)

2 <
(
∑n
i=1

Tri)
2

n
(see

79



A. Alhevaz et.al. / J. Algebra Comb. Discrete Appl. 10(2) (2023) 73–86

[10]), and by the Cauchy-Schwartz inequality we have
(∑n

i=1 Tri

)2
≤ n

∑n
i=1 Tr

2
i , hence for 0 ≤ α ≤

1
2

we get √
2(1 −α)2

∑
1≤i<j≤n(dij)

2 + α2
∑n
i=1 Tr

2
i −

4α2W2(G)
n

n

<

√
(1−α)2(

∑
n
i=1

Tri)
2

n
+ α2

∑n
i=1 Tr

2
i −

α2(
∑
n
i=1

Tri)
2

n

n

=

√
(1 − 2α) (

∑
n
i=1 Tri)

2

n
+ α2

∑n
i=1 Tr

2
i

n

≤
√

(1 − 2α)
∑n
i=1 Tr

2
i + α

2
∑n
i=1 Tr

2
i

n
= (1 −α)

√∑n
i=1 Tr

2
i

n
,

then
√

2ζ
n
≤ (1 −α)

√∑
n
i=1

Tr2
i

n
≤ x ≤

√
ζ.

Now, by Cauchy-Schwarz inequality, we have Tr2i =
(∑n

j=1 dij

)2
≤ n

∑n
j=1 d

2
ij. Hence

∑n
i=1 Tr

2
i ≤

n
∑n
i=1

∑n
j=1 d

2
ij, and then we get

∑
1≤i<j≤n d

2
ij ≥

1
2n

∑n
i=1 Tr

2
i . Then we have√

2(1 −α)2
∑

1≤i<j≤n(dij)
2 + α2

∑n
i=1 Tr

2
i −

4α2W2(G)
n

n

≥

√√√√2(1 −α)2 ∑1≤i<j≤n(dij)2 + α2(∑ni=1 Tri)2n − α2(∑ni=1 Tri)2n
n

=

√
2(1 −α)2

∑
1≤i<j≤n(dij)

2

n
≥

(1 −α)
√∑n

i=1 Tr
2
i

n
.

Hence 0 ≤ y ≤ (1−α)
√∑

n
i=1 Tr

2
i

n
≤
√

2ζ
n
. Therefore

f(x,y) ≤ f

(
(1 −α)

√∑n
i=1 Tr

2
i

n
,

(1 −α)
√∑n

i=1 Tr
2
i

n

)
≤ f

(√
2ζ

n
,

√
2ζ

n

)
.

Consequently,

EDα(G) ≤ (1 −α)
√∑n

i=1 Tr
2
i

n
+

(1 −α)
√∑n

i=1 Tr
2
i

n

+

√
(n− 2)

(
2ζ −

(1 −α)2
∑n
i=1 Tr

2
i

n
−

(1 −α)2
∑n
i=1 Tr

2
i

n2

)
.

The first part is complete.

Suppose the equality holds in (8). From equality in (9), we get√∣∣∣∣∂2 − 2αW(G)n
∣∣∣∣ =

√∣∣∣∣∂3 − 2αW(G)n
∣∣∣∣ = · · · =

√∣∣∣∣∂n−1 − 2αW(G)n
∣∣∣∣,

and hence
n−1∑
i=2

(
∂i −

2αW(G)

n

)2
=

(
2ζ −

(
∂1 −

2αW(G)

n

)2
−
(
∂n −

2αW(G)

n

)2)
.

80



A. Alhevaz et.al. / J. Algebra Comb. Discrete Appl. 10(2) (2023) 73–86

That is,

(n− 2)
(
∂i −

2αW(G)

n

)2
=

2ζ −
(
∂1 −

2αW(G)
n

)2
−
(
∂n −

2αW(G)
n

)2
n− 2

, i = 2, . . . ,n− 1.

Therefore,

∣∣∣∣∂i − 2αW(G)n
∣∣∣∣ =

√√√√2ζ −(∂1 − 2αW(G)n )2 −(∂n − 2αW(G)n )2
n− 2

, i = 2, . . . ,n− 1.

Hence
∣∣∣∂i − 2αW(G)n ∣∣∣ can have at most two distinct values. Thus, if equality holds in (8), then G must

be a connected graph with exactly three or exactly four distinct Dα-eigenvalues.

Next, we consider some lower bounds for the generalized distance energy of graphs. The following
theorem describes a lower bound based on the Wiener index W(G) and the parameter α.

Theorem 2.11. Suppose G is an n-vertex connected graph. Its Wiener index is W(G). We have

EDα(G) >
4αW(G)

n
− 2

√
2(1 −α)2

∑
1≤i<j≤n(dij)

2 + α2
∑n
i=1 Tr

2
i −∂

2
1

n− 1
. (10)

Proof. Let ∂1 ≥ ∂2 ≥ . . . ≥ ∂n be the generalized distance eigenvalues and t is the number of generalized
distance eigenvalues of G which are greater than or equal to 2αW(G)

n
. Note that

2(1 −α)2
∑

1≤i<j≤n

(dij)
2 + α2

n∑
i=1

Tr2i =

n∑
i=1

∂2i ≥ ∂
2
1 + (n− 1)∂

2
n, (11)

hence

∂n ≤

√
2(1 −α)2

∑
1≤i<j≤n(dij)

2 + α2
∑n
i=1 Tr

2
i −∂

2
1

n− 1
.

In view of generalized distance energy, we see that

EDα(G) = 2 max
1≤j≤n

(
j∑
i=1

∂i −
2αjW(G)

n

)
≥ 2

(
n−1∑
i=1

∂i −
2α(n− 1)W(G)

n

)

= 2

(
2αW(G) −∂n −

2α(n− 1)W(G)
n

)
=

4αW(G)

n
− 2∂n

≥
4αW(G)

n
− 2

√
2(1 −α)2

∑
1≤i<j≤n(dij)

2 + α2
∑n
i=1 Tr

2
i −∂

2
1

n− 1
.

By contradiction, we will prove that the inequality in (10) is strict. For this we assume that the equality
in (10) holds true. Therefore, all inequalities above are essentially equalities. Thus we arrive at

max
1≤j≤n

(
j∑
i=1

∂i −
2αjW(G)

n

)
=

n−1∑
i=1

∂i −
2α(n− 1)W(G)

n
. (12)

The equality in (12) holds if and only if t = n−1 and equality occurs in (11) if and only if ∂2 = · · · = ∂n.
In other words, this happens if and only if G has exactly two distinct eigenvalues. Since t = n− 1, we
see that ∂n <

2αW(G)
n

≤ ∂n−1 = ∂n, which is clearly a contradiction. The proof is complete.

81



A. Alhevaz et.al. / J. Algebra Comb. Discrete Appl. 10(2) (2023) 73–86

The following observation is immediate from Lemma 2.2 and Theorem 2.11.

Corollary 2.12. Suppose G is an n-vertex connected graph with minimum transmission degree Trmin
and Wiener index W(G). We have

EDα(G) >
4αW(G)

n
− 2

√
n2(2(1 −α)2

∑
1≤i<j≤n(dij)

2 + α2
∑n
i=1 Tr

2
i ) − 4W2(G)

n− 1
.

Next, we establish the lower bound for EDα(G) involving Wiener index W(G), transmission degrees
as well as generalized distance spectral spread DαS(G).

Theorem 2.13. Suppose that G is an n-vertex connected graph. We have

EDα(G) ≥
2
(

2(1 −α)2
∑

1≤i<j≤n(dij)
2 + α2

∑n
i=1 Tr

2
i −

4α2W2(G)
n

)
DαS(G)

. (13)

Proof. Let ∂1 ≥ ∂2 ≥ . . . ≥ ∂n be the generalized distance eigenvalues of G. Assume x = (xi) and
a = (ai), i = 1, 2, . . . ,n, be real number sequences satisfying

n∑
i=1

|xi| = 1 and
n∑
i=1

xi = 0.

It is proved in [20] that ∣∣∣∣∣
n∑
i=1

aixi

∣∣∣∣∣ ≤ 12
(

max
1≤i≤n

ai − min
1≤i≤n

ai

)
. (14)

Let ai = ∂i −
2αW(G)

n
and xi =

∂i−
2αW(G)

n∑
n
i=1|∂i−2αW(G)n |

, i = 1, 2, . . . ,n. Since

n∑
i=1

xi =

∑n
i=1

(
∂i −

2αW(G)
n

)
∑n
i=1

∣∣∣∂i − 2αW(G)n ∣∣∣ = 0 and
n∑
i=1

|xi| =

∑n
i=1

∣∣∣∂i − 2αW(G)n ∣∣∣∑n
i=1

∣∣∣∂i − 2αW(G)n ∣∣∣ = 1,
the conditions for inequality (14) are satisfied. Hence, we have∣∣∣∣∣∣∣

∑n
i=1

(
∂i −

2αW(G)
n

)2
∑n
i=1

∣∣∣∂i − 2αW(G)n ∣∣∣
∣∣∣∣∣∣∣ ≤

1

2

(
max
1≤i≤n

(
∂i −

2αW(G)

n

)
− min

1≤i≤n

(
∂i −

2αW(G)

n

))
,

that is

2(1 −α)2
∑

1≤i<j≤n(dij)
2 + α2

∑n
i=1 Tr

2
i −

4α2W2(G)
n

EDα(G)
≤

1

2

(
∂1 −

2αW(G)

n
−∂n +

2αW(G)

n

)
,

then, we get

EDα(G) ≥
2
(

2(1 −α)2
∑

1≤i<j≤n(dij)
2 + α2

∑n
i=1 Tr

2
i −

4α2W2(G)
n

)
DαS(G)

.

82



A. Alhevaz et.al. / J. Algebra Comb. Discrete Appl. 10(2) (2023) 73–86

The following bound for DαS(G) was established in [22]:

DαS(G) ≤

√√√√√2

2(1 −α)2 ∑

1≤i<j≤n

(dij)2 + α2
n∑
i=1

Tr2i −
4α2W2(G)

n


. (15)

Hence, using (15) and (13), we obtain the following lower bound for the generalized distance energy.

Corollary 2.14. Suppose that G is an n-vertex connected graph. We have

EDα(G) ≥

√√√√√2

2(1 −α)2 ∑

1≤i<j≤n

(dij)2 + α2
n∑
i=1

Tr2i −
4α2W2(G)

n


.

Corollary 2.15. Assume G is connected and has n vertices. We obtain

EDα(G) ≥ (1 −α)
√

2n(n− 1).

Proof. Since dij ≥ 1 for i 6= j and there are
n(n−1)

2
pairs of vertices in G, from the lower bound of

Corollary 2.14, we get

EDα(G) ≥

√√√√√2

2(1 −α)2 ∑

1≤i<j≤n

(dij)2 + α2
n∑
i=1

Tr2i −α2
n∑
i=1

Tr2i


 ≥ √4(1 −α)2 n(n− 1)

2

= (1 −α)
√

2n(n− 1).

We conclude by giving the following lower bound for the generalized distance energy EDα(G) involv-
ing some graph-theoretic invariants such as W(G), Trmax and Trmin.

Theorem 2.16. Suppose that G is an n-vertex graph. Let Trmax and Trmin be respectively the largest
and the least transmissions of G. Let Γ =

∣∣∣det(Dα(G) − 2αW(G)n I)∣∣∣. We have
EDα(G) ≥ Trmin −

2αW(G)

n
+

√
2ζ −ω2 + (n− 1)(n− 2)

(
Γ

ω

) 2
n−1

, (16)

where 2ζ = 2(1 − α)2
∑

1≤i<j≤n(dij)
2 + α2

∑n
i=1 Tr

2
i −

4α2W2(G)
n

and ω = Trmax −
2αW(G)

n
. Equality

holds if and only if either G is a complete graph or a k-transmission regular graph with three distinct
Dα-eigenvalues (

k,

√
M −k2(1 −α)2

n− 1
+ αk,−

√
M −k2(1 −α)2

n− 1
+ αk

)
,

where M = 2(1 −α)2
∑

1≤i<j≤n(dij)
2.

Proof. It is not difficult to see that

(
EDα(G)

)2
=

n∑
i=1

(
∂i −

2αW(G)

n

)2
+ 2

∑
1≤i<j≤n

∣∣∣∣∂i − 2αW(G)n
∣∣∣∣
∣∣∣∣∂j − 2αW(G)n

∣∣∣∣
= 2ζ + 2

(
∂1 −

2αW(G)

n

) n∑
i=2

∣∣∣∣∂i − 2αW(G)n
∣∣∣∣

+ 2
∑

2≤i<j≤n

∣∣∣∣∂i − 2αW(G)n
∣∣∣∣
∣∣∣∣∂j − 2αW(G)n

∣∣∣∣ , (17)
83



A. Alhevaz et.al. / J. Algebra Comb. Discrete Appl. 10(2) (2023) 73–86

where 2ζ = 2(1−α)2
∑

1≤i<j≤n(dij)
2 +α2

∑n
i=1 Tr

2
i −

4α2W2(G)
n

. It follows from the arithmetic-geometric
mean inequality that

∑
2≤i<j≤n

∣∣∣∣∂i − 2αW(G)n
∣∣∣∣
∣∣∣∣∂j − 2αW(G)n

∣∣∣∣ ≥ (n− 1)(n− 2)2
(

n∏
i=2

∣∣∣∣∂i − 2αW(G)n
∣∣∣∣
) 2
n−1

(18)

=
(n− 1)(n− 2)

2

(
Γ

∂1 −
2αW(G)

n

) 2
n−1

. (19)

Applying the inequality (19), we get from (17) that

(
EDα(G)

)2
≥ 2ζ + 2

(
∂1 −

2αW(G)

n

)(
EDα(G) −

(
∂1 −

2αW(G)

n

))

+ (n− 1)(n− 2)

(
Γ

∂1 −
2αW(G)

n

) 2
n−1

,

which implies that

(
EDα(G)

)2
− 2

(
∂1 −

2αW(G)

n

)
EDα(G) −

[
2ζ − 2

(
∂1 −

2αW(G)

n

)2
+(n− 1)(n− 2)

(
Γ

∂1 −
2αW(G)

n

) 2
n−1 ]

≥ 0.

By solving the above inequality, we get

EDα(G) ≥ ∂1 −
2αW(G)

n
+

√√√√
2ζ −

(
∂1 −

2αW(G)

n

)2
+ (n− 1)(n− 2)

(
Γ

∂1 −
2αW(G)

n

) 2
n−1

.

Applying Corollary 2.4, we have

EDα(G) ≥ Trmin −
2αW(G)

n
(20)

+

√√√√
2ζ −

(
Trmax −

2αW(G)

n

)2
+ (n− 1)(n− 2)

(
Γ

Trmax −
2αW(G)

n

) 2
n−1

.

Hence we get the required result in (16). The first part is complete.

Assume that the equality in (16) holds. The rest inequalities above are essentially equalities. Hence,
∂1 = Trmax = Trmin, and G is a transmission regular graph. It follows from the equality in (18), we see
that

∣∣∣∂2 − 2αW(G)n ∣∣∣ = ∣∣∣∂3 − 2αW(G)n ∣∣∣ = · · · = ∣∣∣∂n − 2αW(G)n ∣∣∣ , and hence
n∑
i=2

(
∂i −

2αW(G)

n

)2
=

(
2ζ −

(
∂1 −

2αW(G)

n

)2)
,

that is,

(n− 1)
(
∂i −

2αW(G)

n

)2
=

(
2ζ −

(
∂1 −

2αW(G)

n

)2)
,

84



A. Alhevaz et.al. / J. Algebra Comb. Discrete Appl. 10(2) (2023) 73–86

then for each i = 2, . . . ,n, we have

∣∣∣∣∂i − 2αW(G)n
∣∣∣∣ =

√√√√2ζ −(∂1 − 2αW(G)n )2
n− 1

.

Consequently,
∣∣∣∂i − 2αW(G)n ∣∣∣ can have at most two distinct values and we arrive at the following:

(i) G has exactly one distinct Dα-eigenvalue. Thus, G = K1.
(ii) G has exactly two distinct Dα-eigenvalues. Using Lemma 2.1, G = Kn.
(iii) G has exactly three distinct Dα-eigenvalues. We have ∂1 = Trmax, and

∣∣∣∂i − 2αW(G)n ∣∣∣ =√
2ζ−(Trmax−

2αW(G)
n )

2

n−1 , i = 2, . . . ,n. Moreover, G is k-transmission regular graph. We have G as a
graph with three distinct Dα-eigenvalues(

k,

√
M −k2(1 −α)2

n− 1
+ αk,−

√
M −k2(1 −α)2

n− 1
+ αk

)
,

in which M = 2(1 −α)2
∑

1≤i<j≤n(dij)
2. Hence the result follows.

Since for any i, we have n− 1 ≤ Tri ≤
n(n−1)

2
, hence from Theorem 2.16, the following observation

is immediate.

Corollary 2.17. Suppose that G is an n-vertex graph and Γ =
∣∣∣det(Dα(G) − 2αW(G)n I)∣∣∣. We have

EDα(G) ≥ n− 1 −
2αW(G)

n
+

√
2ζ −h2 + (n− 1)(n− 2)

(
Γ

h

) 2
n−1

,

where 2ζ = 2(1 −α)2
∑

1≤i<j≤n(dij)
2 + α2

∑n
i=1 Tr

2
i −

4α2W2(G)
n

and h = n(n−1)
2
− 2αW(G)

n
.

Acknowledgements: The authors would like to thank the anonymous referee(s) and the Editor for
their valuable comments to improve the presentation of the manuscript. Y. Shang was supported in part
by the UoA Flexible Fund No. 201920A1001 from Northumbria University.

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	Introduction
	New bounds on generalized distance energy of graphs
	References