# On distance Laplacian energy in terms of graph invariants

Hilal A. Ganiea, Rezwan Ul Shabanb, Bilal A. Ratherc, S. Pirzadad,
aDepartment of School Education, JK Govt. Kashmir, India
b,c,dDepartment of Mathematics, University of Kashmir, Srinagar, India
<EMAIL_ADDRESS><EMAIL_ADDRESS>
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Abstract. For a simple connected graph $G$ of order $n$ having distance
Laplacian eigenvalues
$\rho^{L}_{1}\geq\rho^{L}_{2}\geq\cdots\geq\rho^{L}_{n}$, the distance
Laplacian energy $DLE(G)$ is defined as
$DLE(G)=\sum_{i=1}^{n}\left|\rho^{L}_{i}-\frac{2W(G)}{n}\right|$, where $W(G)$
is the Wiener index of $G$. We obtain a relationship between the Laplacian
energy and distance Laplacian energy for graphs with diameter 2. We obtain
lower bounds for the distance Laplacian energy $DLE(G)$ in terms of the order
$n$, the Wiener index $W(G)$, independence number, vertex connectivity number
and other given parameters. We characterize the extremal graphs attaining
these bounds. We show that the complete bipartite graph has the minimum
distance Laplacian energy among all connected bipartite graphs and complete
split graph has the minimum distance Laplacian energy among all connected
graphs with given independence number. Further, we obtain the distance
Laplacian spectrum of the join of a graph with the union of two other graphs.
We show that the graph $K_{k}\bigtriangledown(K_{t}\cup K_{n-k-t}),1\leq
t\leq\lfloor\frac{n-k}{2}\rfloor$, has the minimum distance Laplacian energy
among all connected graphs with vertex connectivity $k$. We conclude this
paper with a discussion on trace norm of a matrix and the importance of our
results in the theory of trace norm of the matrix
$D^{L}(G)-\frac{2W(G)}{n}I_{n}$.

Keywords: Distance matrix; energy; distance Laplacian matrix; distance
Laplacian energy.

AMS subject classification: 05C50, 05C12, 15A18.

##  1 Introduction

A graph is denoted by $G(V(G),E(G))$, where
$V(G)=\\{v_{1},v_{2},\ldots,v_{n}\\}$ is its vertex set and $E(G)$ is its edge
set. Throughout, $G$ is connected, simple and finite. $|V(G)|=n$ is the order
and $|E(G)|=m$ is the size of $G$. The set of vertices adjacent to $v\in
V(G)$, denoted by $N(v)$, refers to the neighborhood of $v.$ The degree of
$v,$ denoted by $d_{G}(v)$ (we simply write $d_{v}$ if it is clear from the
context) means the cardinality of $N(v)$. A graph is regular if each of its
vertices have the same degree. The adjacency matrix $A=(a_{ij})$ of $G$ is a
$(0,1)$-square matrix of order $n$ whose $(i,j)$-entry is equal to $1$, if
$v_{i}$ is adjacent to $v_{j}$ and equal to $0$, otherwise. Let
$Deg(G)={diag}(d_{1},d_{2},\dots,d_{n})$ be the diagonal matrix of vertex
degrees $d_{i}=d_{G}(v_{i})$, $i=1,2,\dots,n$ associated to $G$. The matrix
$L(G)=Deg(G)-A(G)$ is the Laplacian matrix and its spectrum is the Laplacian
spectrum of $G$. This matrix is real symmetric and positive semi-definite. We
take $0=\mu_{n}\leq\mu_{n-1}\leq\dots\leq\mu_{1}$ to be the Laplacian
eigenvalues of $G$. The Laplacian energy of a graph [10], denoted by $LE(G)$,
is defined as $LE(G)=\sum_{i=1}^{n}|\mu_{i}-\frac{2m}{n}|$. For some recent
papers and related results on Laplacian energy, we refer to [8, 14] and the
references therein. For other undefined notations and terminology, the readers
are referred to [4, 12, 15].
The distance between two vertices $u,v\in V(G),$ denoted by $d_{uv}$, is
defined as the length of a shortest path between $u$ and $v$. The diameter of
$G$ is the maximum distance between any two vertices of $G.$ The distance
matrix of $G$, denoted by $D(G)$, is defined as $D(G)=(d_{uv})_{u,v\in V(G)}$.
The transmission $Tr_{G}(v)$ of a vertex $v$ is defined to be the sum of the
distances from $v$ to all other vertices in $G$, i.e.,
$Tr_{G}(v)=\sum\limits_{u\in V(G)}d_{uv}.$ A graph $G$ is said to be
$k$-transmission regular if $Tr_{G}(v)=k,$ for each $v\in V(G).$ The
transmission number or Wiener index of a graph $G$, denoted by $W(G),$ is the
sum of distances between all unordered pairs of vertices in $G$. Clearly,
$W(G)=\frac{1}{2}\displaystyle\sum_{v\in V(G)}Tr_{G}(v)$. For any vertex
$v_{i}\in V(G)$, the transmission $Tr_{G}(v_{i})$ is called the transmission
degree, shortly denoted by $Tr_{i}$ and the sequence
$\\{Tr_{1},Tr_{2},\ldots,Tr_{n}\\}$ is called the transmission degree sequence
of the graph $G$.
Let $Tr(G)=diag(Tr_{1},Tr_{2},\ldots,Tr_{n})$ be the diagonal matrix of vertex
transmissions of $G$. Aouchiche and Hansen [2] defined the distance Laplacian
matrix of $G$ as $D^{L}(G)=Tr(G)-D(G)$.
Let $\rho^{D}_{1}\geq\rho^{D}_{2}\geq\dots\geq\rho^{D}_{n}$ and
$\rho^{L}_{1}\geq\rho^{L}_{2}\geq\dots\geq\rho^{L}_{n}$ be respectively, the
distance and distance Laplacian eigenvalues of the graph $G$. The distance
energy [11] of $G$ is the sum of the absolute values of the distance
eigenvalues of $G$, that is,
$DE(G)=\displaystyle\sum\limits_{i=1}^{n}|\rho^{D}_{i}|$. For some recent
works on distance energy, we refer to [5, 6, 1] and the references therein.
The distance Laplacian energy $DLE(G)$ [16] of a connected graph $G$ is
defined as

$DLE(G)=\sum\limits_{i=1}^{n}|\rho^{L}_{i}-\frac{2W(G)}{n}|.$

Let $\sigma$ be the largest positive integer such that
$\rho^{L}_{\sigma}\geq\frac{2W(G)}{n}$ and let
$U_{k}^{L}(G)=\displaystyle\sum\limits_{i=1}^{k}\rho^{L}_{i}$ be the sum of
$k$ largest distance Laplacian eigenvalues of $G$. Then using
$\displaystyle\sum\limits_{i=1}^{n}\rho^{L}_{i}=2W(G)$, it can be seen that

$\displaystyle DLE(G)$ $\displaystyle=2\left(U_{\sigma}^{L}(G)-\frac{2\sigma
W(G)}{n}\right)=2\max_{1\leq j\leq
n}\left(\sum\limits_{i=1}^{j}\rho_{i}^{L}(G)-\frac{2jW(G)}{n}\right)$
$\displaystyle=2\max_{1\leq j\leq
n}\left(U_{j}^{L}(G)-\frac{2jW(G)}{n}\right).$

For some recent works on $DLE(G)$, see [5, 6, 7].
The rest of the paper is organized as follows. In Section 2, we obtain a
relationship between the Laplacian energy and distance Laplacian energy for
graphs with diameter $2$. We also obtain a lower bound for the distance
Laplacian energy $DLE(G)$ in terms the order $n$, the Wiener index $W(G)$,
etc. and characterize the extremal graphs attaining these bounds. In Section
3, we study the distance Laplacian energy of connected bipartite graphs and of
connected graphs with given independence number. We show that the complete
bipartite graph has the minimum distance Laplacian energy among all connected
bipartite graphs and complete split graph has the minimum distance Laplacian
energy among all connected graphs with given independence number. In Section
4, we study the distance Laplacian spectrum of join of a graph with the union
of two other graphs. We, also study the distance Laplacian energy of connected
graphs with given vertex connectivity. We show that the graph
$K_{k}\bigtriangledown(K_{t}\cup K_{n-k-t}),1\leq
t\leq\lfloor\frac{n-k}{2}\rfloor$, has the minimum distance Laplacian energy
among all connected graphs with vertex connectivity $k$. We have added a
conclusion at the end to highlight the connection between the distance
Laplacian energy of a connected graph $G$ with the trace norm of the matrix
$D^{L}(G)-\frac{2W(G)}{n}I_{n}$.

##  2 Bounds for the distance Laplacian energy of a graph

We begin with the lemma, which gives the relation between the distance
Laplacian spectrum of a graph and its connected spanning subgraph and can be
found in [2].

###### Lemma 2.1

Let $G$ be a connected graph of order $n$ and size $m$, where $m\geq n$ and
let $G^{{}^{\prime}}=G-e$ be a connected graph obtained from $G$ by deleting
an edge. Let $\rho_{1}^{L}(G)\geq\rho_{2}^{L}(G)\geq\cdots\geq\rho_{n}^{L}(G)$
and
$\rho_{1}^{L}(G^{{}^{\prime}})\geq\rho_{2}^{L}(G^{{}^{\prime}})\geq\cdots\geq\rho_{n}^{L}(G^{{}^{\prime}})$
be respectively, the distance Laplacian eigenvalues of $G$ and
$G^{{}^{\prime}}$. Then $\rho_{i}^{L}(G^{{}^{\prime}})\geq\rho_{i}^{L}(G)$
holds for all $1\leq i\leq n.$

The following lemma shows that the distance Laplacian eigenvalues of a
connected graph $G$ of diameter $2$ are completely determined by the Laplacian
eigenvalues of the graph $G$ and can be found in [2].

###### Lemma 2.2

Let $G$ be a connected graph of order $n\geq 2$ having diameter $d=2$. Let
$\mu_{1}\geq\mu_{2}\geq\cdots\geq\mu_{n-1}>\mu_{n}=0$ be the Laplacian
eigenvalues and
$\rho^{L}_{1}(G)\geq\rho^{L}_{2}(G)\geq\cdots\geq\rho^{L}_{n-1}(G)>\rho^{L}_{n}(G)=0$
be the distance Laplacian eigenvalues of $G$. Then
$\rho^{L}_{i}(G)=2n-\mu_{n-i}$, for $i=1,2,\dots,n-1$.

A well known fact is that the complete graph $K_{n}$ is the only graph of
diameter $1$. So for this graph, clearly $DLE(K_{n})=2(n-1)$. Therefore, in
the rest of the paper we will be dealing with graphs of diameter greater or
equal to $2$. The following result gives the distance Laplacian energy of a
graph of diameter $2$ in terms of sum of the Laplacian eigenvalues of $G$.

###### Theorem 2.3

Let $G$ be a connected graph of order $n\geq 3$ and size $m$ having diameter
$2$. Then

$\displaystyle
DLE(G)=2\Big{(}\sigma\Big{(}\frac{2m}{n}+2\Big{)}-2m+S_{n-\sigma-1}(G)\Big{)},$

where $S_{n-\sigma-1}(G)=\sum\limits_{i=1}^{n-\sigma-1}\mu_{i}$ is the sum of
the $n-\sigma-1$ largest Laplacian eigenvalues of $G$ and
$\sigma,1\leq\sigma\leq n-2$, is the number of distance Laplacian eigenvalues
of $G$ which are greater than or equal to $\frac{2W(G)}{n}$.

Proof. Let $G$ be a connected graph of order $n$ having $m$ edges. Since
diameter of $G$ is two, it follows that
$Tr(v_{i})=d_{i}+2(n-1-d_{i})=2n-2-d_{i}$, for all $v_{i}\in V(G)$ and so
$2W(G)=2n(n-1)-2m$. Let $\sigma$ be the number of distance Laplacian
eigenvalues of $G$ which are greater than or equal to $\frac{2W(G)}{n}$. Using
Lemma 2.2 and the definition of Laplacian energy, we have

$\displaystyle DLE(G)$
$\displaystyle=2\left(\sum\limits_{i=1}^{\sigma}\rho_{i}^{L}(G)-\frac{2\sigma
W(G)}{n}\right)$
$\displaystyle=2\left(\sum\limits_{i=1}^{\sigma}(2n-\mu_{n-i})-\sigma(2n-2-\frac{2m}{n})\right)$
$\displaystyle=2\left(\sigma\big{(}\frac{2m}{n}+2\big{)}-\sum\limits_{i=1}^{\sigma}\mu_{n-i}\right)=2\Big{(}\sigma\big{(}\frac{2m}{n}+2\big{)}-2m+S_{n-\sigma-1}(G)\Big{)},$

where $S_{n-\sigma-1}(G)=\sum\limits_{i=1}^{n-\sigma-1}\mu_{i}$ is the sum of
the $n-\sigma-1$ largest Laplacian eigenvalues of $G$.

From Theorem 2.3, it is clear that any lower bound or upper bound for the
graph invariant $S_{k}(G)$, the sum of $k$ largest Laplacian eigenvalues of
$G$, gives a lower bound or upper bound for $DLE(G)$. In fact, there is a
conjecture by A. Brouwer for the graph invariant $S_{k}(G)$, which is stated
as follows.

###### Conjecture 1

If $G$ is any graph with order $n$ and size $m$, then

$S_{k}(G)=\sum\limits_{i=1}^{k}\mu_{i}\leq m+\binom{k+1}{2},\qquad\text{for
any}\qquad k\in\\{1,2,\dots,n\\}.$

This conjecture has been shown to be true for various families of graphs, but
as a whole this conjecture is still open. For some recent developments on
Brouwer’s conjecture, we refer to [9] and the references therein.

The following theorem gives a relation between the distance Laplacian energy
and Laplacian energy of a graph of diameter $2$.

###### Theorem 2.4

Let $G$ be a connected graph of order $n\geq 3$ and size $m$ having diameter
$2$. Let $\sigma$ and $t$, $1\leq\sigma,t\leq n-2$, be respectively the number
of distance Laplacian eigenvalues and number of Laplacian eigenvalues of $G$
which are greater than or equal to $\frac{2W(G)}{n}$ and $\frac{2m}{n}$. Then

$\displaystyle LE(G)-2\Big{(}\frac{2m}{n}-2(n-1)+2t\Big{)}\leq DLE(G)\leq
LE(G)+4\Big{(}\sigma-\frac{m}{n}\Big{)}.$

Proof. Let $\sigma$ be the number of distance Laplacian eigenvalues of $G$
which are greater than or equal to $\frac{2W(G)}{n}$. Then by definition of
distance Laplacian energy, we have

$\displaystyle DLE(G)=2\Big{(}U^{L}_{\sigma}(G)-\frac{2\sigma
W(G)}{n}\Big{)}=2\max_{1\leq j\leq
n-1}\Big{(}U^{L}_{j}(G)-\frac{2jW(G)}{n}\Big{)}.$ (2.1)

Also, if $t$ is the number of Laplacian eigenvalues of $G$ which are greater
than or equal to $\frac{2m}{n}$, then by definition of Laplacian energy, we
have

$\displaystyle LE(G)=2\Big{(}S_{t}(G)-\frac{2tm}{n}\Big{)}=2\max_{1\leq j\leq
n-1}\Big{(}S_{j}(G)-\frac{2jm}{n}\Big{)}.$ (2.2)

Using Theorem 2.3 and second equality of (2.2), we have

$\displaystyle DLE(G)=$ $\displaystyle
2\Big{(}\sigma\big{(}\frac{2m}{n}+2\big{)}-2m+S_{n-\sigma-1}(G)\Big{)}$
$\displaystyle=2\Big{(}2\sigma-\frac{2m}{n}\Big{)}+2\Big{(}S_{n-\sigma-1}(G)-\frac{2m(n-\sigma-1)}{n}\Big{)}$
$\displaystyle\leq LE(G)+4\Big{(}\sigma-\frac{m}{n}\Big{)},$

as $1\leq n-\sigma-1\leq n-1$. Using Lemma 2.2 and (2.1), we have

$\displaystyle LE(G)=$ $\displaystyle
2\Big{(}S_{t}(G)-\frac{2tm}{n}\Big{)}=2\Big{(}(2n-\frac{2m}{n})t-\sum\limits_{i=1}^{t}\rho^{L}_{n-i}\Big{)}$
$\displaystyle=2\Big{(}(2n-\frac{2m}{n})t-2W(G)+\sum\limits_{i=1}^{n-t-1}\rho^{L}_{i}\Big{)}$
$\displaystyle=2\Big{(}\frac{2m}{n}-2(n-1)+2t\Big{)}+2\Big{(}\sum\limits_{i=1}^{n-t-1}\rho^{L}_{i}-\frac{2(n-t-1)W(G)}{n}\Big{)}$
$\displaystyle\leq 2\Big{(}\frac{2m}{n}-2(n-1)+2t\Big{)}+DLE(G),$

as $2W(G)=2n(n-1)-2m$ and $1\leq n-t-1\leq n-1$. This completes the proof.

From Theorem 2.4, the following observation is immediate.

###### Corollary 2.5

Let $G$ be a connected graph of order $n\geq 3$ and size $m$ having diameter
$2$. Let $\sigma$ and $t$, $1\leq\sigma,t\leq n-2$, be respectively the number
of distance Laplacian eigenvalues and number of Laplacian eigenvalues of $G$
which are greater than or equal to $\frac{2W(G)}{n}$ and $\frac{2m}{n}$. Then
$LE(G)>DLE(G)$, provided that $m>\sigma n$ and $LE(G)<DLE(G)$, provided that
$t<n-\frac{m}{n}-2$.

For the star graph $K_{1,n-1}$ the Laplacian spectrum is $\\{n,1^{[n-2]},0\\}$
and $m=n-1$. It is easy to see that $t=1$ for $K_{1,n-1}$ and so the
inequality $LE(K_{1,n-1})<DLE(K_{1,n-1})$ is valid for $n\geq 4$.

The following observation also follows from Theorem 2.4.

###### Corollary 2.6

Let $G$ be a connected graph of order $n\geq 3$ and size $m$ having diameter
$2$. Let $\sigma$ and $t$, $1\leq\sigma,t\leq n-2$, be respectively the number
of distance Laplacian eigenvalues and number of Laplacian eigenvalues of $G$
which are greater than $\frac{2W(G)}{n}$ and $\frac{2m}{n}$. Then $\sigma\geq
n-(t+1)$.

The following lemma [5] gives an upper bound for the second smallest distance
Laplacian eigenvalue $\rho^{L}_{n-1}(G)$ in terms of order $n$ and the minimum
transmission degree $Tr_{\min}$ of the graph $G$.

###### Lemma 2.7

Let $G$ be a connected graph of order $n\geq 3$ having minimum transmission
$Tr_{\min}$ and second smallest distance Laplacian eigenvalue
$\rho^{L}_{n-1}(G)$. Then

$\displaystyle\rho^{L}_{n-1}(G)\leq\frac{n}{n-1}Tr_{\min},$ (2.3)

with equality if and only if $G$ is a graph containing a vertex of
transmission degree $n-1$.

Now, we obtain a lower bound for the distance Laplacian energy in terms the
order $n$, the Wiener index $W(G)$ and the minimum transmission $Tr_{\min}$ of
the graph $G$.

###### Theorem 2.8

Let $G$ be a connected graph of order $n\geq 3$ having minimum transmission
degree $Tr_{\min}$ and Wiener index $W(G)$. Then

$\displaystyle DLE(G)\geq\frac{8W(G)}{n}-\frac{2nTr_{\min}}{n-1},$ (2.4)

equality occurs if and only if $\sigma(G)=n-2$ and $G$ containing a vertex of
transmission degree $n-1$.

Proof. Let $G$ be a connected graph of order $n\geq 3$ having distance
Laplacian eigenvalues
$\rho^{L}_{1}(G)\geq\rho^{L}_{2}(G)\geq\cdots\geq\rho^{L}_{n-1}(G)\geq\rho^{L}_{n}(G)=0$.
Let $\sigma$ be the number of distance Laplacian eigenvalues of $G$ which are
greater than or equal to $\frac{2W(G)}{n}$. Using
$\sum\limits_{i=1}^{n}\rho^{L}_{i}(G)=2W(G)$ and the definition of Laplacian
energy [5], we have

$\displaystyle DLE(G)$ $\displaystyle=2\max_{1\leq j\leq
n-1}\left(\sum\limits_{i=1}^{j}\rho_{i}^{L}(G)-\frac{2jW(G)}{n}\right)$
$\displaystyle\geq
2\left(\sum\limits_{i=1}^{n-2}\rho^{L}_{i}(G)-\frac{2(n-2)W(G)}{n}\right)$
$\displaystyle=2\left(2W(G)-\rho^{L}_{n-1}(G)-\frac{2(n-2)W(G)}{n}\right)=\frac{8W(G)}{n}-2\rho^{L}_{n-1}(G)$
$\displaystyle\geq\frac{8W(G)}{n}-\frac{2nTr_{\min}}{n-1}.$

Equality occurs in (2.4) if and only if equality occurs in

$\displaystyle\max_{1\leq j\leq
n-1}\left(\sum\limits_{i=1}^{j}\rho_{i}^{L}(G)-\frac{2jW(G)}{n}\right)=\left(\sum\limits_{i=1}^{n-2}\rho^{L}_{i}(G)-\frac{2(n-2)W(G)}{n}\right)$
(2.5)

and equality occurs in $\rho^{L}_{n-1}(G)\leq\frac{n}{n-1}Tr_{\min}$. Equality
occurs in (2.5) if and only if $\sigma=n-2$ and by Lemma 2.7 equality occurs
in $\rho^{L}_{n-1}(G)\leq\frac{n}{n-1}Tr_{\min}$ if and only if $G$ contains a
vertex having transmission degree $n-1$. This shows that Equality occurs in
(2.4) if and only if $\sigma=n-2$ and $G$ contains a vertex having
transmission degree $n-1$. This completes the proof.

The parameter $t$ gives the number of Laplacian eigenvalues of a graph $G$
which are in the interval $[\frac{2m}{n},n]$. This parameter has been studied
for various families of graphs and is presently as active topic of research in
the field of spectral theory of graphs. Like the parameter $t$, the parameter
$\sigma$ is concerned with the distribution of distance Laplacian eigenvalues
of a connected graph $G$. In fact, the value of $\sigma$ gives the number of
distance Laplacian eigenvalues which are in the interval
$[\frac{2W(G)}{n},\infty)$. It will be of interest to discuss the following
problem for the parameter $\sigma$.
Problem 1. Characterize the graphs having $\sigma=1$, $\sigma=n-2$ and
$\sigma=n-1$. Establish relations between $\sigma$ with different parameters
of the graph $G$.

For the transmission regular graphs $G$, it can be seen that
$\sigma=n-\gamma$, where $\gamma$ is the positive inertia of the distance
matrix of the graph $G$.

##  3 Distance Laplacian energy of bipartite graphs and graphs with given
independence number

In this section, we show that among all connected bipartite graphs the
complete bipartite graph has the minimum distance Laplacian energy. We also
show that among all connected graphs with independence number $t$, the
complete split graph has the minimum distance Laplacian energy.

Among all connected bipartite graphs, the complete bipartite graph has the
minimum distance Laplacian energy can be seen as follows.

###### Theorem 3.1

Let $G$ be a connected bipartite graph of order $n\geq 3$ with partite sets of
cardinality $a$ and $b$, $a+b=n$.
(1). If $a<b$ or $a>b$, then

$\displaystyle
DLE(G)\geq\left\\{\begin{array}[]{lr}4n^{2}-6n-4ab-\frac{4(n-2)W(G)}{n},&\mbox{if
$2ab\geq n(b-2)\ or\ 2ab\geq n(a-2)$},\\\
2(b-1)\left(2n-a-\dfrac{2W(G)}{n}\right),&\mbox{if $2ab<n(b-2),~{}a<b$},\\\
2(a-1)\left(2n-b-\dfrac{2W(G)}{n}\right),&\mbox{if
$2ab<n(a-2),~{}a>b$},\end{array}\right.$

with equality if and only if $G\cong K_{a,b}$.
(2). If $n=2a$ and $n\geq 5,$ then $DLE(G)\geq
12a(a-1)-\frac{4(n-2)W(G)}{n}=12b(b-1)-\frac{4(n-2)W(G)}{n}$. Equality occurs
if and only if $G\cong K_{a,a}.$

Proof. Let $G$ be a connected bipartite graph of order $n$ with vertex set
$V(G).$ Let $V(G)=V_{1}\cup V_{2},$ with $|V_{1}|=a,|V_{2}|=b$ be a
bipartition of the vertex set $V(G)$ of $G$. Since $G$ is a connected
bipartite graph with partite sets of cardinality $a$ and $b$, it follows that
$G$ is a spanning subgraph of the complete bipartite graph $K_{a,b}$.
Therefore, by Lemma 2.1, we have
$\partial_{i}^{L}(G)\geq\partial_{i}^{L}(K_{a,b})$, for all $i=1,2,\dots,n$.
With this and the definition of distance Laplacian energy $DLE(G)$ , we have

$\displaystyle DLE(G)$
$\displaystyle=2\left(\sum\limits_{i=1}^{\sigma}\rho_{i}^{L}(G)-\frac{2\sigma
W(G)}{n}\right)=2\max_{1\leq j\leq
n}\left(\sum\limits_{i=1}^{j}\rho_{i}^{L}(G)-\frac{2jW(G)}{n}\right)$
$\displaystyle\geq 2\max_{1\leq j\leq
n}\left(\sum\limits_{i=1}^{j}\rho_{i}^{L}(K_{a,b})-\frac{2jW(G)}{n}\right),$
(3.6)

where $\sigma$ is the largest positive integer such that
$\rho_{\sigma}^{L}(G)\geq\frac{2W(G)}{n}.$
In [3], it can be seen that the distance Laplacian spectrum of $K_{a,b}$ is
$\\{(2n-a)^{[b-1]},(2n-b)^{[a-1]},n,0\\}$ and $2W(K_{a,b})=2n^{2}-2n-2ab$.
Since $a$ and $b$ are positive integers, therefore by trichotomy law, we have
$a<b$ or $a>b$ or $a=b$. We first prove the result for $a<b$. If $a<b$, then
$2n-a\geq 2n-b$. Also $2n-a>n$ holds as $n>a$. Likewise, if $n>b$, then
$2n-b>n$. These observations imply that $2n-a$ is the distance Laplacian
spectral radius of $K_{a,b}$ in case of $a<b$ and so we always have
$2n-a\geq\frac{2W(K_{a,b})}{n}$. The case $a>b$ can be discussed similarly.
For the eigenvalue $n$, we have $n<\frac{2W(K_{a,b})}{n}$ implying that
$n^{2}-2n-2ab>0$, which further gives

$\displaystyle 2a^{2}-2an+n^{2}-2n>0,\quad\text{as}\quad a+b=n.$ (3.7)

Consider the polynomial $f(a)=2a^{2}-2an+n^{2}-2n$, for $1\leq a<n$. The
discriminant of this polynomial is $d=4n(4-n)$. Clearly for $n>4$, we have
$d<0$ and so for this $n$, we always have $f(a)>0$. For $n=3,4$, it can be
seen by direct calculation that inequality (3.7) holds. This shows that
$n<\frac{2W(K_{a,b})}{n}$ holds for all $a<b$. For the eigenvalue $2n-b,$ we
have $2n-b\geq\frac{2W(K_{a,b})}{n}$ implying that $2ab\geq n(b-2)$. This
shows that if $2ab\geq n(b-2)$, then $\frac{2W(G)}{n}\leq 2n-b$ and if
$2ab<n(b-2)$, we have $\frac{2W(G)}{n}>2n-b.$ In other words, if $2ab\geq
n(b-2)$, then $\sigma=n-2$ while if $2ab<n(b-2)$, then $\sigma=b-1$.
Therefore, we have the following cases to consider.
Case (i). If $2ab\geq n(b-2)$, then the number of eigenvalues which are
greater or equal to $\frac{2W(G)}{n}$ are $n-2$, that is $\sigma=n-2.$
$Since$$1\leq\sigma\leq n-1$, from (3) it follows that

$\displaystyle DLE(G)$ $\displaystyle\geq
2\left(\sum\limits_{i=1}^{n-2}\rho_{i}^{L}(K_{a,b})-\frac{2(n-2)W(G)}{n}\right)$
$\displaystyle=2\left((b-1)(2n-a)+(a-1)(2n-b)-\frac{2(n-2)W(G)}{n}\right)$
$\displaystyle=4n^{2}-6n-4ab-\frac{4(n-2)W(G)}{n}.$

Case(ii). If $2ab<n(b-2)$, then the number of eigenvalues, which are greater
or equal to $\frac{2W(G)}{n}$, is $b-1$,that is $\sigma=b-1.$ Since
$1\leq\sigma\leq n-1$, from (3) it follows that

$\displaystyle DLE(G)$ $\displaystyle\geq
2\left(\sum\limits_{i=1}^{b-1}\rho_{i}^{L}(K_{a,b})-\frac{2(b-1)W(G)}{n}\right)$
$\displaystyle=2(b-1)\left(2n-a-\dfrac{2W(G)}{n}\right).$

If $a>b$, then proceeding similarly as above, it can be seen that $2n-b$ is
the distance Laplacian spectral radius, $n<\frac{2W(K_{a,b})}{n}$ holds for
all $a,b$, while as $\frac{2W(G)}{n}\leq 2n-a$, for $2ab\geq n(a-2)$ and
$\frac{2W(G)}{n}>2n-a,$ for $2ab<n(a-2)$. Using this information in (3), we
get

$\displaystyle DLE(G)\geq 4n^{2}-6n-4ab-\frac{4(n-2)W(G)}{n},$

for $2ab\geq n(a-2)$ and

$\displaystyle DLE(G)\geq 2(a-1)\left(2n-b-\dfrac{2W(G)}{n}\right),$

for $2ab<n(a-2)$. Equality occurs in each of the inequalities above if and
only if equality occurs in (3). It is clear that equality occurs in (3) if and
only if $G\cong K_{a,b}$. This gives that equality occurs if and only if
$G\cong K_{a,b}$. This completes the proof for these cases.
If $a=b$, then $n=2a$ and so the distance Laplacian spectrum of $K_{a,a}$ is
$\\{(2n-a)^{[n-2]},n,0\\}$ and
$\dfrac{2W(K_{a,a})}{n}=\dfrac{2n^{2}-an-3n+2a}{n}.$ Since
$2n-a\geq\dfrac{2W(K_{a,a})}{n}$ holds for all $a$ and for $n\geq 5$, we have
$\dfrac{2W(K_{a,a})}{n}>n$, it follows that $\sigma=n-2$. Therefore, from (3),
we have

$\displaystyle DLE(G)$ $\displaystyle\geq
2\left(\sum\limits_{i=1}^{n-2}\rho_{i}^{L}(K_{a,b})-\frac{2(n-2)W(G)}{2}\right)$
$\displaystyle=12a(a-1)-\frac{4(n-2)W(G)}{n}.$

Equality case can be discussed similarly as above. This completes the proof.

For the complete bipartite graph $K_{a,b}$ with $a+b=n$ and $n\geq 3$, using
Theorem 3.1, we have the following:
For $2ab\geq n(b-2)$ with $a<b$; or $2ab\geq n(a-2)$ with $a>b$, we have
$4n^{2}-6n-4ab-\frac{4(n-2)W(K_{a,b})}{n}>12a(a-1)-\frac{4(n-2)W(K_{a,b})}{n}$
implying that $8a^{2}+(4n-12)a-(4n^{2}-6n)<0$. The zeros of the polynomial
$f(a)=8a^{2}+(4n-12)a-(4n^{2}-6n)$ are $-n+\frac{3}{2},\frac{n}{2}$. This
implies that $f(a)<0$, for all $a\in(-n+\frac{3}{2},\frac{n}{2})$. This shows
that $4n^{2}-6n-4ab-\frac{4(n-2)W(K_{a,b})}{n}>12a(a-1)-\frac{4(n-2)W(G)}{n}$,
holds for all $a<\frac{n}{2}$. For $2ab<n(b-2),~{}a<b$, we have
$2(b-1)\left(2n-a-\dfrac{2W(K_{a,b})}{n}\right)>12a(b-1)-\frac{4(2b-2)W(K_{a,b})}{n}$
giving that $2n-a+\dfrac{2W(K_{a,b})}{n}>6a$. Using
$2W(K_{a,b})=2n^{2}-2n-2an+2a^{2}=(n-1)^{2}+(n-a)^{2}+a^{2}-1$, we get
$2a^{2}-9an+4n^{2}-2n>0$. The zeros of the polynomial
$g(a)=2a^{2}-9an+4n^{2}-2n$ are
$y_{1}=\frac{9n+\sqrt{49n^{2}+16n}}{4},y_{2}=\frac{9n-\sqrt{49n^{2}+16n}}{4}$.
This shows that $g(a)>0$, for all $a>y_{1}$ and for all $a<y_{2}$. Since
$y_{1}>n$, it follows that $g(a)>0$, for all $a<y_{2}$. For $n\geq 3$, it is
easy to see that $y_{2}>\frac{n-1}{2}$, implying that $g(a)>0$, for all
$a<\frac{n-1}{2}$. This shows that
$2(b-1)\left(2n-a-\dfrac{2W(K_{a,b})}{n}\right)>12a(b-1)-\frac{4(2b-2)W(K_{a,b})}{n}$,
holds for all $a<\frac{n-1}{2}$. For $2ab<n(a-2),~{}a>b$, proceeding similarly
as above it can be seen that
$2(a-1)\left(2n-b-\dfrac{2W(K_{a,b})}{n}\right)>12b(a-1)-\frac{4(2a-2)W(K_{a,b})}{n}$,
holds for all $b<\frac{n-1}{2}$. Now, using Theorem 3.1 and the fact that $a$
and $b=n-a$ are positive integers, we have the following observation.

###### Corollary 3.2

Among all bipartite graphs of order $n\geq 3$, the complete bipartite graph
$K_{\lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil}$ has the minimum
distance Laplacian energy.

The next observation follows from Lemma 2.1 and Theorem 3.1.

###### Corollary 3.3

Let $G$ be a connected bipartite graph with partite sets of cardinality $a$
and $b$, $a+b=n$. Let
$\rho_{1}^{L}(G)\geq\rho_{2}^{L}(G)\cdots\geq\rho_{n-1}^{L}(G)>\rho_{n}^{L}(G)=0$
be the distance Laplacian eigenvalues of $G$. Then $\rho_{i}^{L}(G)\geq 2n-a,$
for all $1\leq i\leq b-1$, $\rho_{i}^{L}(G)\geq 2n-b,$ for all $b\leq i\leq
a+b-2$, $\rho_{n-1}^{L}(G)\geq n$. Equality occurs in each of these
inequalities if and only if $G\cong K_{a,b}$.

A complete split graph, denoted by $CS_{t,n-t}$, is the graph consisting of a
clique on $t$ vertices and an independent set (a subset of vertices of a graph
is said to be an independent set if the subgraph induced by them is an empty
graph) on the remaining $n-t$ vertices, such that each vertex of the clique is
adjacent to every vertex of the independent set.

The following theorem shows that among all connected graphs with given
independence number $n-t$, $1\leq t\leq n-1$, the complete split graph
$CS_{t,n-t}$ has the minimum distance Laplacian energy.

###### Theorem 3.4

Let $G$ be a connected graph of order $n\geq 3$ having independence number
$n-t$, $1\leq t\leq n-1$. Then

$\displaystyle
DLE(G)\geq\left\\{\begin{array}[]{lr}2(n-t-1)\Big{(}2n-t-\frac{2W(G)}{n}\Big{)},&\mbox{if
$t<n-\frac{1}{2}-\sqrt{n+\frac{1}{4}}$},\\\
4n^{2}-2nt-4n+2t(t+1)-\frac{4(n-1)W(G)}{n},&\mbox{if $t\geq
n-\frac{1}{2}-\sqrt{n+\frac{1}{4}}$},\end{array}\right.$

equality occurs in each case if and only if $G\cong CS_{t,n-t},~{}1\leq t\leq
n-1$.

Proof. Let $G$ be a connected graph of order $n\geq 3$ having independence
number $n-t$. Let $CS_{t,n-t}$ be the complete split graph having independence
number $n-t$. It is clear that $G$ is a spanning subgraph of $CS_{t,n-t}$.
Therefore, by Lemma 2.1, we have
$\rho_{i}^{L}(G)\geq\rho_{i}^{L}(CS_{t,n-t})$. Let $\sigma$ be the largest
positive integer such that $\rho_{\sigma}^{L}(G)\geq\frac{2W(G)}{n}.$ With
this information, it follows that

$\displaystyle DLE(G)$
$\displaystyle=2\left(\sum\limits_{i=1}^{\sigma}\rho_{i}^{L}(G)-\frac{2\sigma
W(G)}{n}\right)=2\max_{1\leq j\leq
n}\left(\sum\limits_{i=1}^{j}\rho_{i}^{L}(G)-\frac{2jW(G)}{n}\right)$
$\displaystyle\geq 2\max_{1\leq j\leq
n}\left(\sum\limits_{i=1}^{j}\rho_{i}^{L}(CS_{t,n-t})-\frac{2jW(G)}{n}\right).$
(3.8)

The distance Laplacian spectrum [3] of the complete split graph $CS_{t,n-t}$
is $\\{(2n-t)^{[n-t-1]},n^{[t]},0\\}$ with
$\frac{2W(CS_{t,n-t})}{n}=\frac{2n(n-t-1)+t(t+1)}{n}$. Since $n-t\geq 1$, it
follows that $2n-t$ is the distance Laplacian spectral radius of $CS_{t,n-t}$.
For the eigenvalue $n$, we have
$n<\frac{2W(CS_{t,n-t})}{n}=\frac{2n(n-t-1)+t(t+1)}{n}$, which after
simplification gives

$\displaystyle t^{2}-(2n-1)t+n^{2}-2n>0.$ (3.9)

Consider the polynomial $f(t)=t^{2}-(2n-1)t+n^{2}-2n$, for $1\leq t\leq n-1$.
The roots of this polynomial are $x_{1}=n-\frac{1}{2}-\sqrt{n+\frac{1}{4}}$
and $x_{2}=n-\frac{1}{2}+\sqrt{n+\frac{1}{4}}$. This implies that $f(t)>0$ for
all $t<x_{1}$ and $f(t)>0$ for all $t>x_{2}$. Since $x_{2}>n$ and $t\leq n-1$,
it follows that inequality (3.9) holds for all
$t<n-\frac{1}{2}-\sqrt{n+\frac{1}{4}}$. From this, it follows that for
$t<n-\frac{1}{2}-\sqrt{n+\frac{1}{4}}$, we have $\sigma=n-t-1$ and for $t\geq
n-\frac{1}{2}-\sqrt{n+\frac{1}{4}}$, we have $\sigma=n-1$. Therefore, we have
the following cases to consider.
Case (i). If $t<n-\frac{1}{2}-\sqrt{n+\frac{1}{4}}$, then $\sigma=n-t-1$.
Since $1\leq\sigma\leq n-1$, from (3) it follows that

$\displaystyle DLE(G)$ $\displaystyle\geq 2\max_{1\leq j\leq
n}\left(\sum\limits_{i=1}^{j}\rho_{i}^{L}(CS_{t,n-t})-\frac{2jW(G)}{n}\right)$
$\displaystyle\geq
2\left(\sum\limits_{i=1}^{n-t-1}\rho_{i}^{L}(CS_{t,n-t})-\frac{2(n-t-1)W(G)}{n}\right)$
$\displaystyle=2(n-t-1)\Big{(}2n-t-\frac{2W(G)}{n}\Big{)}.$

Case (ii). If $t\geq n-\frac{1}{2}-\sqrt{n+\frac{1}{4}}$, then $\sigma=n-1$.
Since $1\leq\sigma\leq n-1$, from (3) it follows that

$\displaystyle DLE(G)$ $\displaystyle\geq 2\max_{1\leq j\leq
n}\left(\sum\limits_{i=1}^{j}\rho_{i}^{L}(CS_{t,n-t})-\frac{2jW(G)}{n}\right)$
$\displaystyle\geq
2\left(\sum\limits_{i=1}^{n-1}\rho_{i}^{L}(CS_{t,n-t})-\frac{2(n-1)W(G)}{n}\right)$
$\displaystyle=2(2n-t)(n-t-1)+2nt-\frac{4(n-1)W(G)}{n}$
$\displaystyle=4n^{2}-2nt-4n+2t(t+1)-\frac{4(n-1)W(G)}{n}.$

Equality occurs in all the inequalities above if and only if equality occurs
in (3). It is clear that equality occurs in (3) if and only if $G\cong
CS_{t,n-t}$. This shows that equality occurs in all the inequalities above if
and only if $G\cong CS_{t,n-t}$. This completes the proof.

We characterized the extremal graphs which attain the minimum value for the
distance Laplacian energy among all connected bipartite graphs and among all
connected graphs with given independence number. The following problems will
be of interest for the future research.
Problem 2. Characterize the extremal graphs which attain the maximum value for
the distance Laplacian energy among all connected bipartite graphs of order
$n$.
Problem 3. Characterize the extremal graphs which attain the maximum value for
the distance Laplacian energy among all connected graphs of order $n$ with
independence number $\alpha$.

##  4 Distance Laplacian energy of graphs with given connectivity

In this section, we obtain the distance Laplacian spectrum of the join of a
connected graph $G_{0}$ with the union of two connected graphs $G_{1}$ and
$G_{2}$. We show the existence of some new families of graphs having all the
distance Laplacian eigenvalues integers. We also determine the graph with the
minimum distance Laplacian energy among all the connected graphs with given
vertex connectivity.

The vertex connectivity of a graph $G$, denoted by $\kappa(G)$, is the minimum
number of vertices of $G$ whose deletion disconnects $G$. Let
$\mathcal{F}_{n}$ be the family of simple connected graphs on $n$ vertices and
let $\mathcal{V}_{n}^{k}=\\{G\in\mathcal{F}_{n}:\kappa(G)\leq k\\}$.

Let $G_{1}(V_{1},E_{1})$ and $G_{2}(V_{2},E_{2})$ be two graphs on disjoint
vertex sets $V_{1}$ and $V_{2}$ of order $n_{1}$ and $n_{2}$, respectively.
The union of graphs $G_{1}$ and $G_{2}$ is the graph $G_{1}\cup
G_{2}=(V_{1}\cup V_{2},E_{1}\cup E_{2}).$ The join of graphs $G_{1}$ and
$G_{2}$, denoted by $G_{1}\bigtriangledown G_{2}$ is the graph consisting of
$G_{1}\cup G_{2}$ and all edges joining the vertices in $V_{1}$ and the
vertices in $V_{2}.$ In other words, the join of two graphs $G_{1}$ and
$G_{2},$ denoted by $G_{1}\bigtriangledown G_{2},$ is the graph obtained from
$G_{1}$ and $G_{2}$ by joining each vertex of $G_{1}$ to every vertex of
$G_{2}.$

In the following theorem, we find the distance Laplacian spectrum of the join
of a connected graph $G_{0}$ with the union of two connected graphs $G_{1}$
and $G_{2}$ in terms of the distance Laplacian spectrum of the graphs
$G_{0},G_{1}$ and $G_{2}$.

###### Theorem 4.1

Let $G_{0},G_{1}$ and $G_{2}$ be the connected graphs of order $n_{0},n_{1}$
and $n_{2}$ respectively. The distance Laplacian spectrum of
$G_{0}\bigtriangledown(G_{1}\cup G_{2})$ consists of eigenvalues
$(\lambda+n_{1}+n_{2}),(\mu+n_{0}+2n_{2}),(\zeta+n_{0}+2n_{1}),2n-n_{0},n,0$,
where $n=n_{0}+n_{1}+n_{2}$ and $\lambda,\mu,\zeta$ varies over the non-zero
distance Laplacian eigenvalues of $G_{0},G_{1}$, $G_{2}$, respectively.

Proof. Let $G_{i}$, $i=0,1,2$ be the connected graphs of order $n_{i}$. Let
$G=G_{0}\bigtriangledown(G_{1}\cup G_{2})$ be the join of graphs $G_{0}$ and
$G_{1}\cup G_{2}$. Clearly, $G$ is a graph of diameter $2.$ Let
$D^{L}(G_{0}),D^{L}(G_{1})$ and $D^{L}(G_{2})$ be respectively the distance
Laplacian matrices of the graphs $G_{0},G_{1}$ and $G_{2}$. By suitably
labelling the vertices of $G,$ it can be seen that the distance Laplacian
matrix of $G$ is

$D^{L}(G)=\begin{pmatrix}b_{0}&-J_{n_{0}\times n_{1}}&-J_{n_{0}\times
n_{2}}\\\ -J_{n_{1}\times n_{0}}&b_{1}&-2J_{n_{1}\times n_{2}}\\\
-J_{n_{2}\times n_{0}}&-2J_{n_{2}\times n_{1}}&b_{2}\end{pmatrix},$

where
$b_{0}=(2n-n_{1}-n_{2})I_{n_{0}}-2J_{n_{0}}-D^{L}(G_{0}),b_{i}=(2n-n_{0})I_{n_{i}}-2J_{n_{i}}-D^{L}(G_{i})$,
for $i=1,2$, $J_{n_{i}\times n_{j}}$ is the all one matrix of order
$n_{i}\times n_{j}$, $I_{n_{k}}$ is the identity matrix of order $n_{k}$. Let
$e_{n_{i}}=(1,1,\dots,1)$ be the all $1-$vector of order $n_{i},i=0,1,2.$ It
is well known that $e_{n_{i}}$ is an eigenvector of $G_{i}$ for the distance
Laplacian eigenvalue $0$ and any other eigenvector $x$ of $G_{i}$ is
orthogonal to $e_{n_{i}}$. Let $x$ be any eigenvector of $G_{0}$ for the non-
zero distance Laplacian eigenvalue $\lambda$. Then $x\perp e_{n_{0}}.$
Consider the column vector $X=(x^{T}\ \ \ \ 0^{T}_{n_{1}\times 1}\ \ \ \
0^{T}_{n_{2}\times 1})^{T}$, We have

$D^{L}(G)X=\begin{pmatrix}(2n-n_{1}-n_{2})I_{n_{0}}-2J_{n_{0}}-D^{L}(G_{0})x\\\
0\\\ 0\end{pmatrix}=(2n-n_{1}-n_{2}-\lambda)X$

giving that $2n-n_{1}-n_{2}-\lambda$ is an eigenvalue of $D^{L}(G)$ for each
non-zero distance Laplacian eigenvalue $\lambda$ of $G_{0}.$ In this way, we
get $n_{0}-1$ distance Laplacian eigenvalues of $G.$ Similarly, if $0\neq\mu$
is a distance Laplacian eigenvalue of $G_{1}$ with eigenvector $y$, $y\perp
e_{n_{1}}$, then it can be seen that the column vector $Y=(0^{T}_{n_{0}\times
1}\ \ \ y^{T}\ \ \ 0^{T}_{n_{2}\times 1})^{T}$ is an eigenvector of $D^{L}(G)$
for the eigenvalue $2n-n_{0}-\mu$. This implies that $2n-n_{0}-\mu$ is an
eigenvalue of $D^{L}(G)$ for each non-zero distance Laplacian eigenvalue $\mu$
of $G_{1}.$ In this way, we get other $n_{1}-1$ distance Laplacian eigenvalues
of $G.$ Lastly, if $0\neq\zeta$ is a distance Laplacian eigenvalue of $G_{2}$
with eigenvector $z,z\perp e_{n_{2}}$, then it can be seen that the column
vector $Z=(0^{T}_{n_{0}\times 1}\ \ \ 0^{T}_{n_{1}\times 1}\ \ \ z^{T})^{T}$
is an eigenvector of $D^{L}(G)$ for the eigenvalue $2n-n_{0}-\zeta$. This
implies that $2n-n_{0}-\zeta$ is an eigenvalue of $D^{L}(G)$ for each non-zero
distance Laplacian eigenvalue $\zeta$ of $G_{2}.$ In this way, we get another
$n_{2}-1$ distance Laplacian eigenvalues of $G.$ Thus, we get a total of
$n_{0}-1+n_{1}-1+n_{2}-1=n-3$ distance Laplacian eigenvalues of $G$. The
remaining three distance Laplacian eigenvalues of $G$ are given by the
quotient matrix of $D^{L}(G)$, which is

$M=\begin{pmatrix}n_{1}+n_{2}&-n_{1}&-n_{2}\\\ -n_{0}&n_{0}+2n_{2}&-2n_{2}\\\
-n_{0}&-2n_{1}&n_{0}+2n_{1}\end{pmatrix}.$

Since each row sum of matrix $M$ is zero, it follows that one of the
eigenvalues of this matrix is $0$. The other two eigenvalues of $M$ are given
by the roots of $x^{2}-(3n-n_{0})x+n(2n-n_{0})=0$. This completes the proof.

A graph $G$ is said to an adjacency integral(Laplacian integral, distance
integral) graph if all its adjacency eigenvalues(Laplacian eigenvalues,
distance eigenvalues) are integers. Likewise, a graph $G$ is called distance
Laplacian integral graph if all its distance Laplacian eigenvalues are
integers. From Theorem 4.1 , it is clear that if all the graphs $G_{0}$,
$G_{1}$ and $G_{2}$ are distance Laplacian integral, then the graph
$G=G_{0}\bigtriangledown(G_{1}\cup G_{2})$ is also a distance Laplacian
integral graph. It is well known that the complete graph $K_{n}$, the complete
bipartite graph $K_{a,n-a}$, the complete split graph $CS_{a,n-a}$, the
pineapple graph $PA_{n,p}$ (the graph obtained from a clique $K_{n-p}$ by
adding $p$ pendent edges to a vertex of $K_{n-p}$) and the graph $S^{+}$(the
graph obtained from the star graph $K_{1,n-1}$ by adding an edge between two
pendent vertices) are all Laplacian integral graphs. Therefore, using Theorem
4.1, we have the following observation.

###### Theorem 4.2

Let $n=n_{0}+n_{1}+n_{2}$, Then each of the graphs
$K_{n_{0}}\bigtriangledown(K_{n_{1}}\cup K_{n_{2}})$,
$K_{n_{0}}\bigtriangledown(K_{n_{1}-a,a}\cup K_{n_{2}})$,
$K_{n_{0}-a,a}\bigtriangledown(K_{n_{1}}\cup K_{n_{2}})$,
$K_{n_{0}-a,a}\bigtriangledown(K_{n_{1}-b,b}\cup K_{n_{2}})$,
$K_{n_{0}-a,a}\bigtriangledown(K_{n_{1}-b,b}\cup K_{n_{2}-c,c})$,
$CS_{a,n_{0}-a}\bigtriangledown(K_{n_{1}}\cup K_{n_{2}})$,
$CS_{a,n_{0}-a}\bigtriangledown(CS_{b,n_{1}-b}\cup K_{n_{2}})$,
$CS_{a,n_{0}-a}\bigtriangledown(CS_{b,n_{1}-b}\cup CS_{c,n_{2}-c}$,
$K_{n_{0}-a,a}\bigtriangledown(CS_{b,n_{1}-b}\cup K_{n_{2}})$,
$PA_{n_{0},p}\bigtriangledown(K_{n_{1}}\cup K_{n_{2}})$,
$CS_{a,n_{0}-a}\bigtriangledown(PA_{n_{1},p}\cup K_{n_{2}})$,
$CS_{a,n_{0}-a}\bigtriangledown(PA_{n_{1},p}\cup K_{n_{2}-c,c})$, etc are
distance Laplacian integral graphs.

The following theorem shows that among all connected graphs with given vertex
connectivity $k$ the graph $K_{k}\bigtriangledown(K_{t}\cup K_{n-k-t})$ has
the minimum distance Laplacian energy.

###### Theorem 4.3

Let $G\in\mathcal{V}_{n}^{k}$ be a connected graph of order $n\geq 4$ having
vertex connectivity number $k$. Then $DLE(G)\geq
t(2n-k-t+1)-\frac{2tW(G)}{n}$, for $1\leq k<\frac{n-2t}{2}-\frac{n}{2t}$;
$DLE(G)\geq t(3n-2k-2t)+n(n-t-k-1)-\frac{2(n-k-1)W(G)}{n}$, for
$\frac{n-2t}{2}-\frac{n}{2t}\leq k<n-t-\frac{n}{2t}$ and $DLE(G)\geq
n(n-1)+2t(n-k-t)-\frac{2(n-1)W(G)}{n}$, for $n-t-\frac{n}{2t}\leq k\leq n-2$;
if $k=n-1$, then $G\cong K_{n}$ and so $DLE(G)=2n-2$. Equality occurs in each
of these inequalities if and only if $G\cong K_{k}\bigtriangledown(K_{t}\cup
K_{n-k-t}),1\leq t\leq\lfloor\frac{n-k}{2}\rfloor$.

Proof. Let $G$ be a connected graph of order $n$ with vertex connectivity $k$,
$1\leq k\leq n-1$. We first show that $U^{L}_{i}(G)\geq
U^{L}_{i}(K_{k}\bigtriangledown(K_{t}\cup K_{n-t-k}))$, for all
$i=1,2,\dots,n$. If $k=n-1$, then $G\cong K_{n}$ and
$K_{k}\bigtriangledown(K_{t}\cup K_{n-t-k})=K_{n}$ and so equality holds in
this case. Assume that $1\leq k\leq n-2$, that is, $G$ is not a complete
graph. Suppose that $G$ is the connected graph of order $n$ with vertex
connectivity $k$ for which the spectral parameter $U^{L}_{i}(G)$ has the
minimum possible value. Then, it is clear that $G\in\mathcal{V}_{n}^{k}$ and
$U^{L}_{i}(G)$ attains the minimum value for $G$. Let $S$ be a vertex cut set
of $G$ with $|S|=k$. Let $G_{1},G_{2},\dots,G_{r}$ be the connected components
of the graph $G-S$. We will show the number of components of graph $G-S$ is
two, that is $r=2$. For if, $r>2$, then adding an edge between any two
components, say $G_{1}$ and $G_{2}$ of $G-S$ gives the graph
$G^{{}^{\prime}}=G+e$, which is such that the vertex connectivity of
$G^{{}^{\prime}}$ is $k$. Clearly, $G^{{}^{\prime}}\in\mathcal{V}_{n}^{k}$,
also by Lemma 2.1, we have $U^{L}_{i}(G)>U^{L}_{i}(G^{{}^{\prime}})$. This is
a contradiction to the fact $U^{L}_{i}(G)$ attains the minimum possible value
for $G$. Therefore, we must have $r=2$. Further, we claim that each of the
components $G_{1}$, $G_{2}$ and the vertex induced subgraph $\langle S\rangle$
are cliques. For if one among them say $G_{1}$ is not a clique, then adding an
edge between the two non adjacent vertices of $G_{1}$ gives a graph $H$ having
vertex connectivity same as the vertex connectivity of $G$. Clearly,
$H\in\mathcal{V}_{n}^{k}$ and by Lemma 2.1, we have
$U^{L}_{i}(G)>U^{L}_{i}(H)$, Which is a contradiction as $U^{L}_{i}(G)$
attains the minimum possible value for $G$. Thus, $G$ must be of the form
$G=K_{k}\bigtriangledown(K_{t}\cup K_{n-k-t}),1\leq
t\leq\lfloor\frac{n-k}{2}\rfloor$. This shows that for all
$G\in\mathcal{V}_{n}^{k}$, the spectral parameter $U^{L}_{i}(G)$ has the
minimum possible value for the graph $K_{k}\bigtriangledown(K_{t}\cup
K_{n-k-t})$. That is, for all $G\in\mathcal{V}_{n}^{k}$, we have
$U^{L}_{i}(G)\geq U^{L}_{i}(K_{k}\bigtriangledown(K_{t}\cup K_{n-t-k})).$ With
this, from the definition of distance Laplacian energy, it follows that

$\displaystyle DLE(G)$ $\displaystyle=2\left(U_{\sigma}^{L}(G)-\frac{2\sigma
W(G)}{n}\right)=2\max_{1\leq j\leq
n}\left(\sum\limits_{i=1}^{j}\rho_{i}^{L}(G)-\frac{2jW(G)}{n}\right)$
$\displaystyle\geq 2\max_{1\leq j\leq
n}\left(\sum\limits_{i=1}^{j}\rho_{i}^{L}(K_{k}\bigtriangledown(K_{t}\cup
K_{n-t-k}))-\frac{2jW(G)}{n}\right).$ (4.10)

Taking $G_{n_{0}}=K_{k},G_{n_{1}}=K_{t},G_{n_{2}}=K_{n-k-t},n_{0}=k,n_{1}=t$
and $n_{2}=n-t-k$ in Theorem 4.1, we find that the distance Laplacian spectrum
of the graph $K_{k}\bigtriangledown(K_{t}\cup K_{n-t-k})$ is
$\\{2n-k,(2n-t-k)^{[t-1]},(n+t)^{[n-t-k-1]},n^{[k]},0\\}$. Let $\sigma$ be the
number of distance Laplacian eigenvalues of $K_{k}\bigtriangledown(K_{t}\cup
K_{n-t-k})$ which are greater than or equal to that
$\frac{2W(K_{k}\bigtriangledown(K_{t}\cup
K_{n-t-k}))}{n}=\frac{n^{2}-n+2nt-2t^{2}-2kt}{n}.$ It is easy to see that
$2n-k$ is the distance Laplacian spectral radius of the graph
$K_{k}\bigtriangledown(K_{t}\cup K_{n-t-k})$ and so for this eigenvalue, we
always have $2n-k\geq\frac{2W(K_{k}\bigtriangledown(K_{t}\cup
K_{n-t-k}))}{n}$. For the eigenvalue $2n-k-t$, we have
$2n-k-t\geq\frac{2W(K_{k}\bigtriangledown(K_{t}\cup
K_{n-t-k}))}{n}=\frac{n^{2}-n+2nt-2t^{2}-2kt}{n}$ giving that

$\displaystyle 2t^{2}-(3n-2k)t+(n^{2}+n-kn)\geq 0.$ (4.11)

The zeros of the polynomial $g_{1}(t)=2t^{2}-(3n-2k)t+(n^{2}+n-kn)$ are
$y_{1}=\frac{3n-2k+\sqrt{(n-2k)^{2}-8n}}{2}$ and
$y_{2}=\frac{3n-2k-\sqrt{(n-2k)^{2}-8n}}{2}$. This shows that $g_{1}(t)\geq
0$, for all $t\leq y_{2}$ and $t\geq y_{1}$. Since,
$\frac{n-k}{2}<\frac{3n-2k-\sqrt{(n-2k)^{2}-8n}}{2}=y_{2}$ always holds, it
follows that $g_{1}(t)\geq 0$, for all $t\leq\frac{n-k}{2}$. For the
eigenvalue $n+t$, we have $n+t\geq\frac{2W(K_{k}\bigtriangledown(K_{t}\cup
K_{n-t-k}))}{n}=\frac{n^{2}-n+2nt-2t^{2}-2kt}{n}$ giving that
$2t^{2}-(n-2k)t+n\geq 0$, which in turn gives
$k\geq\frac{n-2t}{2}-\frac{n}{2t}$. This shows that
$n+t\geq\frac{2W(K_{k}\bigtriangledown(K_{t}\cup K_{n-t-k}))}{n}$, for all
$k\geq\frac{n-2t}{2}-\frac{n}{2t}$ and
$n+t<\frac{2W(K_{k}\bigtriangledown(K_{t}\cup K_{n-t-k}))}{n}$, for all
$k<\frac{n-2t}{2}-\frac{n}{2t}$. Lastly, for the eigenvalue $n$, we have
$n\geq\frac{2W(K_{k}\bigtriangledown(K_{t}\cup
K_{n-t-k}))}{n}=\frac{n^{2}-n+2nt-2t^{2}-2kt}{n}$ giving that
$2t^{2}-(2n-2k)t+n\geq 0$, which in turn gives that $k\geq n-t-\frac{n}{2t}$.
This shows that $n\geq\frac{2W(K_{k}\bigtriangledown(K_{t}\cup
K_{n-t-k}))}{n}$, for all $k\geq n-t-\frac{n}{2t}$ and
$n<\frac{2W(K_{k}\bigtriangledown(K_{t}\cup K_{n-t-k}))}{n}$, for all
$k<n-t-\frac{n}{2t}$. From this discussion it follows that, if $1\leq
k<\frac{n-2t}{2}-\frac{n}{2t}$, then $\sigma=t$, if
$\frac{n-2t}{2}-\frac{n}{2t}\leq k<n-t-\frac{n}{2t}$, then $\sigma=n-k-1$ and
if $k\geq n-t-\frac{n}{2t}$, then $\sigma=n-1$. If $1\leq
k<\frac{n-2t}{2}-\frac{n}{2t}$, then from (4), it follows that

$\displaystyle DLE(G)$ $\displaystyle\geq 2\max_{1\leq j\leq
n}\left(\sum\limits_{i=1}^{j}\rho_{i}^{L}(K_{k}\bigtriangledown(K_{t}\cup
K_{n-t-k}))-\frac{2jW(G)}{n}\right)$ $\displaystyle\geq
2\left(\sum\limits_{i=1}^{t}\rho_{i}^{L}(K_{k}\bigtriangledown(K_{t}\cup
K_{n-t-k}))-\frac{2tW(G)}{n}\right)$
$\displaystyle=t(2n-k-t+1)-\frac{2tW(G)}{n}.$

If $\frac{n-2t}{2}-\frac{n}{2t}\leq k<n-t-\frac{n}{2t}$, then from (4), it
follows that

$\displaystyle DLE(G)$ $\displaystyle\geq 2\max_{1\leq j\leq
n}\left(\sum\limits_{i=1}^{j}\rho_{i}^{L}(K_{k}\bigtriangledown(K_{t}\cup
K_{n-t-k}))-\frac{2jW(G)}{n}\right)$ $\displaystyle\geq
2\left(\sum\limits_{i=1}^{n-k-1}\rho_{i}^{L}(K_{k}\bigtriangledown(K_{t}\cup
K_{n-t-k}))-\frac{2(n-k-1)W(G)}{n}\right)$
$\displaystyle=t(3n-2k-2t)+n(n-t-k-1)-\frac{2(n-k-1)W(G)}{n}.$

If $n-t-\frac{n}{2t}\leq k\leq n-1$, then from (4), it follows that

$\displaystyle DLE(G)$ $\displaystyle\geq 2\max_{1\leq j\leq
n}\left(\sum\limits_{i=1}^{j}\rho_{i}^{L}(K_{k}\bigtriangledown(K_{t}\cup
K_{n-t-k}))-\frac{2jW(G)}{n}\right)$ $\displaystyle\geq
2\left(\sum\limits_{i=1}^{n-1}\rho_{i}^{L}(K_{k}\bigtriangledown(K_{t}\cup
K_{n-t-k}))-\frac{2(n-1)W(G)}{n}\right)$
$\displaystyle=n(n-1)+2t(n-k-t)-\frac{2(n-1)W(G)}{n}.$

This completes the proof.

The next observation follows from the proof of the Theorem 4.3.

###### Corollary 4.4

Let $G$ be a connected graph of order $n\geq 4$ having vertex connectivity
$\kappa\leq k$. Let
$\rho_{1}^{L}(G)\geq\rho_{2}^{L}(G)\geq\cdots\geq\rho_{n-1}^{L}(G)>\rho_{n}^{L}(G)=0$
be the distance Laplacian eigenvalues of $G$. Then $\rho_{1}^{L}(G)\geq 2n-k,$
$\rho_{i}^{L}(G)\geq 2n-t-k$, for $2\leq i\leq t$, $\rho_{i}^{L}(G)\geq n+t$,
for $t+1\leq i\leq n-k-1$ and $\rho_{i}^{L}(G)\geq n$, for $n-k\leq i\leq
n-1$. Equality occurs in each of these inequalities if and only if $G\cong
K_{k}\bigtriangledown(K_{t}\cup K_{n-t-k}).$

In the later part of this section, we characterized the extremal graphs which
attain the minimum value for the distance Laplacian energy among all connected
graphs with given vertex connectivity. The following problem will be of
interest for the future research.
Problem 4. Characterize the extremal graphs which attain the maximum value for
the distance Laplacian energy among all connected graphs of order $n$ with
given vertex connectivity.

##  5 Conclusion

Let $\mathbb{M}_{n}(\mathbb{C})$ be the set of all square matrices of order
$n$ with complex entries. The trace norm of a matrix
$M\in\mathbb{M}_{n}(\mathbb{C})$ is defined as $\lVert
M\rVert_{*}=\sum\limits_{i=1}^{n}\sigma_{i}(M)$, where
$\sigma_{1}(M)\geq\sigma_{2}(M)\geq\cdots\geq\sigma_{n}(M)$ are the singular
values of $M$(i.e. the square roots of the eigenvalues of $MM^{*}$, where
$M^{*}$ is the complex conjugate of $M$). It is well known that for a
symmetric matrix $M$, if $\sigma_{i}(M)$ the singular values and
$\lambda_{i}(M)$, $i=1,2,\dots,n,$ are the eigenvalues, then
$\sigma_{i}(M)=|\lambda_{i}(M)|$. In the light of this definition, it follows
that the distance Laplacian energy $DLE(G)$ of a connected graph $G$ is the
trace norm of the matrix $D^{L}(G)-\frac{2W(G)}{n}I_{n}$, where $I_{n}$ is the
identity matrix of order $n$. It is an interesting problem in Matrix theory to
determine among a given class of matrices the matrix (or the matrices) which
attain the maximum value and the minimum value for the trace norm. The trace
norm of matrices associated with the graphs and digraphs are extensively
studied. For some recent papers in this direction see [13] and the references
therein.
Therefore, in this language Theorem 2.4 gives relation between the trace norm
of the matrices $D^{L}(G)-\frac{2W(G)}{n}I_{n}$ and $L(G)-\frac{2m}{n}I_{n}$,
when $G$ is a connected graph of diameter two; Theorem 2.8 gives a lower bound
for the trace norm of $D^{L}(G)-\frac{2W(G)}{n}I_{n}$ in terms of the order
and the trace of the matrix $D^{L}(G)$; Theorem 3.1 gives that among all
bipartite connected graphs $G$, the complete bipartite $K_{a,b}$ attains the
minimum trace norm for the matrix $D^{L}(G)-\frac{2W(G)}{n}I_{n}$; Theorem 3.4
gives that among all connected graphs $G$ with given independence number
$n-t$, $1\leq t\leq n-1$, the complete split graph $CS_{t,n-t}$ attains the
minimum trace norm for the matrix $D^{L}(G)-\frac{2W(G)}{n}I_{n}$ and Theorem
4.3 gives that among all connected graphs $G$ with given vertex connectivity
$k$ the graph $K_{k}\bigtriangledown(K_{t}\cup K_{n-k-t})$ attains the minimum
trace norm for the matrix $D^{L}(G)-\frac{2W(G)}{n}I_{n}$.

Acknowledgements. The research of S. Pirzada is supported by SERB-DST, New
Delhi under the research project number MTR/2017/000084.

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