1. Introduction
Let G be a connected simple graph. The node (also referred to as vertex) set of graph G is and the edge set of G is represented by . Denote by the order, i.e., the number of nodes, and the size, i.e., the number of edges in G. For a node , the neighborhood of v, denoted by , consists of all nodes adjacent to it. The number of nodes in is called the degree of v and is signified by or simply if the underlying graph G is obvious. A regular graph has all its degrees equal. The complement of the graph G, denoted by , is the graph with the same vertex set with G but whose edge set consists of the edges not present in G. As usual, , , , and denote, respectively, the complete graph on n vertices, the complete bipartite graph on vertices, the path on n vertices, and the cycle on n vertices. The adjacency matrix is a -matrix with -entry being 1, if is adjacent to and 0 otherwise. The diagonal matrix containing degrees () is denoted by . The Laplacian matrix and signless Laplacian matrix are defined, respectively, by and . Their eigenvalues (also referred to as spectra) are often referred to as Laplacian spectrum and signless Laplacian spectrum, respectively. These two matrices are semi-definite, and the spectra of them are sorted, respectively, as and .
Given two nodes
, the distance between them is denoted by
, which is the number of edges along the shortest path between them. The maximum distance between any pair of nodes is called the diameter of
G.
is the distance matrix. Various spectral properties of
have been reported in, e.g., the survey by Aouchiche and Hansen [
1]. Recall that a cut vertex (cut edge) in a connected graph is one whose deletion increases its number of connected components. A vertex-cut set (respectively, edge-cut set) of a connected graph
G is a set
S of vertices (respectively, a set
L of edges) whose removal disconnects
G (the number of connected components of
(respectively,
) is greater than one). Moreover, a cut ideal of a graph defined in the literature to record the relations between cuts of the graph. There is a relationship between the notions of cut set and cut ideal associated with a graph with the distance matrix of a connected graph. For some literature regarding this relationship, we refer to [
2,
3,
4,
5].
Given
,
is called the transmission of
v. It is the sum of distances from all nodes to
v. If, for all
,
, then the graph
G is
k-transmission regular. The well known Wiener index (also referred to as transmission) of
G represents the sum of all individual transmissions, namely
.
is also called the transmission degree and is written as
for simplicity. Accordingly, the transmission degree sequence is denoted by
. Define
as the second transmission degree of node
and
as the diagonal matrix of node transmissions of
G. Recent works [
6,
7,
8,
9] explored the spectral properties of (signless) Laplacian of the distance matrix in connected graphs. Given
G, its distance Laplacian and distance signless Laplacian are, respectively, defined as
and
. Many results have been reported on the spectral properties of
and
including spectral radius, second largest eigenvalue, and the smallest eigenvalue. For some recent works, we refer to [
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20] (and the references therein).
Recently, in [
21], Cui et al. introduced the generalized distance matrix
as a convex combinations of
and
. Namely,
, where
. Since
, and
, any result regarding the spectral properties of generalized distance matrix has its counterpart for each of these particular graph matrices, and these counterparts follow immediately from a single proof. In fact, this matrix reduces to merging the distance spectral and distance (signless) Laplacian spectral theories. Since the eigenvalues of
are real, we can arrange them as
.
The notion of graph energy has its origin in theoretical chemistry and was put forward by Gutman [
22]. Let
be the adjacency eigenvalues of
G. The graph energy of
G is defined as
[
23]. Since its inception, graph energy has been extensively studied across chemical and mathematical literature. The upper and lower bounds for
have been reported in some pioneering work (e.g., [
24,
25,
26]). The work by Gutman [
22] signposted a fruitful relationship between graph energy and various eigenvalues of matrices associated to graphs. This leads to a wealth of work on energy-like graph spectral invariant in regards to, e.g., (signless) Laplacian matrix [
27,
28], Randić matrix [
29], and distance matrix [
30]. More recent results can be found in [
17,
18,
24,
29,
30,
31,
32,
33]. In another direction, the graph energy has been extended to digraphs and various energies of digraphs such as energy [
34] and skew energy [
35] were put forward and extensively studied. This concept was generalized by Nikiforov by defining the energy of any matrix [
31]. Many studies, both theory- and application-oriented, have been done along this direction [
24,
32,
34]. For example, if
are the eigenvalues of a molecular matrix
M, the expression
was proposed in [
36] to facilitate the design of quantitative structure for different types of organic compounds.
Let
,
, and
be, respectively, the distance eigenvalues, distance Laplacian eigenvalues, and distance signless Laplacian eigenvalues of graph
G. The distance energy of a graph
G was defined as the sum of the absolute values of the distance eigenvalues of
G [
30]:
The following fundamental relationships hold
For more results on distance energy, we refer the reader to [
16]. Analogous to the definition of (signless) Laplacian energy, the concept of distance (signless) Laplacian energy was explored in [
12,
16,
37]. The distance Laplacian energy of a graph
G is defined as the mean deviation of the values of the distance Laplacian eigenvalues of
G:
Similarly, the distance signless Laplacian energy can be viewed as the mean deviation of the values of the distance signless Laplacian eigenvalues of
G:
Recent results on
and
can be found (see, e.g., [
12,
16,
38]).
Motivated by the above works, we here introduce an energy-like quantity based on the generalized distance matrix
. We define the auxiliary eigenvalues
, corresponding to the generalized distance eigenvalues, as
Analogous to the definitions of distance Laplacian energy
and distance signless Laplacian energy
, the generalized distance energy of
G can be thought of as the mean deviation of the values of the generalized distance eigenvalues of
G, namely,
Therefore, we have
In addition, since
and
thanks to the definition of
we see that
where
Moreover, and . This implies that the concept of generalized distance energy of a graph G merges the theories of distance energy and distance signless Laplacian energy. Hence, it is desirable to tackle the quantity and its related structural and algebraic properties including upper and lower bounds, and the parameter (i.e., ) dependency on the graph topology.
The remainder of the paper is organized as follows. Some preliminary results and useful lemmas are presented in
Section 2. The main results, namely the upper bounds and lower bounds for the generalized distance energy
are presented, respectively, in
Section 3 and
Section 4. The bounds link important graph parameters such as the Wiener index
and the transmission degrees. The extremal graph attaining these bounds are also characterized. We show that the generalized distance energy
can be obtained from the corresponding distance energy
for the transmission regular graphs. Among other results, we prove that, if
G is a connected graph of order
and
, then
where the equality holds if and only if
. It is revealed from this result that the complete graphs attain the minimum generalized distance energy over all connected graphs. We also prove that, if
T is a tree of order
and
, then
where the equality holds if and only if
. It is revealed from this result that the minimum generalized distance energy within trees are attained by the star graphs. We conclude in
Section 5.
4. Lower Bounds for the Generalized Distance Energy
In this section, some new lower bounds for the generalized distance energy of a connected graph G are presented. These bounds connects some interesting graph parameters such as the Wiener index and the transmission degrees. As in the case of upper bounds, the extremal graphs are derived. The complete graphs are shown to have the minimal generalized distance energy among all connected graphs, while the star graphs have the minimal generalized distance energy among all trees.
The following result gives a lower bound for generalized distance energy through the Wiener index .
Theorem 5. Let G be a connected graph of order and let . Then,where the equality holds if and only if G is a transmission regular graph with one positive distance eigenvalue. Proof. Let
G be a connected graph of order
n. Suppose that
t is the integer such that
and
Then, by the definition of generalized distance energy,
We show that
where
Let
s be any positive integer such that
For
we have
For
that is,
This shows that, for any value of
s we have
then
which gives the result in Equation (
9). Thanks to Lemma 3,
Assuming that equality occurs in Equation (
8), then equality occurs in Lemma 3 and
. Since equality occurs in Lemma 3 if and only if
G is a transmission regular graph, we see that that equality occurs in Equation (
8) if and only if
G is a transmission regular graph with
. Let
G be a
k-transmission regular graph having distance eigenvalues
. Then, by Theorem 2, we have
and
, for
. Since
, it follows that
, which gives
, in turn giving
as
. This implies that the equality holds in Equation (
8) if and only if
G is a transmission regular graph with a single positive distance eigenvalue. □
From Theorem 5, it is clear that for a fixed value of the parameter , among connected graphs G of order , the transmission regular graphs with a single positive distance eigenvalue has the minimum generalized distance energy .
Since, for a connected graph G, we have , for all , it then follows that , where the equality holds if and only if . The following corollary is an immediate result.
Corollary 1. Let G be a connected graph of order and let . Then,where the equality holds if and only if . Proof. The proof follows from above observation by using Theorem 5 and the fact that is a transmission regular graph with a single positive distance eigenvalue. □
From Corollary 1, it is clear that, for a fixed value of the parameter
, among all the connected graphs
G of order
, the complete graph
has the minimum generalized distance energy
. We note that the conclusion given by Corollary 1 is in agree with the results known for the distance energy obtained by putting
[
1] and for the distance signless Laplacian energy obtained by putting
[
16].
For a connected graph G of order n and size m, we have , for all . It follows that , with equality if and only if G is a graph of diameter at most 2.
Corollary 2. Let G be a connected graph of order having m edges and let . Then,where the equality holds if and only if or G is the cocktail party graph, G is a regular line graph of diameter two, or G is a regular exceptional graph of diameter two. Proof. Using above observation in Theorem 5 the inequality follows. Equality occurs if and only if
G is a transmission regular graph of diameter at most 2 having one positive distance eigenvalue. Since a graph of diameter at most 2 is transmission regular if and only if it is regular, it follows that equality occurs if and only if
G is a regular graph of diameter at most 2 having one positive distance eigenvalue. Now, using the discussion before the Corollary 1 of [
45], the equality case follows. □
Corollary 2 implies that the minimum value of the generalized distance energy is amongst all connected graphs of order n with m edges. This value is attained by the complete graph , the cocktail party graph, a regular line graph of diameter 2, and a regular exceptional graph of diameter 2.
The following result gives a lower bound for generalized distance energy . The Wiener index and the transmission degrees are involved.
Theorem 6. Let G be a connected graph of order and let . Then,In the case of , the equality hold if and only if G is a transmission regular graph with a single positive distance eigenvalue. Proof. It can be proved similarly by following Lemma 2 and the line of arguments in the proof of the Theorem 5. □
Remark 1. Since all distance regular graphs are transmission regular graphs, by Lemma 9, equality occurs in Theorems 5 and 6 for each of the graphs: a Doob graph, a double odd graph, a Johnson graph, a cocktail party graph, a polygon, a halved cube, a Hamming graph, the Petersen graph, the Schläfli graph, the Gosset graph, the dodecahedron, the icosahedron, and one of the three Chang graphs. It would be inspiring to identify all transmission regular graphs (which are not distance regular) having a single positive distance eigenvalue.
Next, we present a lower bound for the generalized distance energy of a trees via the Wiener index for .
Theorem 7. Let T be a tree of order and let . Then,where the equality holds if and only if Proof. From the equality in Equation (
9), we have
Suppose that the equality holds in Equation (
10). Then, equality occurs in Lemma 1 and
. Equality occurs in Lemma 1 if and only if
. For the graph
, the generalized distance spectrum is
, where
and
. It is now easy to see that
for the graph
. Thus, it follows that equality occurs in Equation (
10) if and only if
. □
For the star graph , we have . Therefore, for any tree T of order n, we have . The following corollary is an immediate result of Theorem 7.
Corollary 3. Let T be a tree of order and let . Then,where the equality holds if and only if By Corollary 3, it is clear that, for fixed
, star graph
has the minimum generalized distance energy among all the trees of order
. We note that the conclusion given by Corollary 3 is in agree with the results known for the distance energy obtained by putting
[
1] and for the distance signless Laplacian energy obtained by putting
[
16].
The following result presents another lower bound for the generalized distance energy again via the Wiener index .
Theorem 8. Let G be connected graph of order n and let . Then,where the equality holds if and only if and . Proof. Let
G be a connected graph of order
n having generalized distance eigenvalues
. As shown in Theorem 5,
where
t is the largest positive integer satisfying
and
. Since
and
G is a spanning subgraph of
, it follows from Lemma 4 that
, for all
. This gives that
as the generalized distance spectrum of
is
. Using Equation (
12) in Equation (
11), we get
Assuming that equality occurs, then there is equality in Equation (
12). Since the equality holds in Equation (
12) if and only if
and
, it is equivalent to
and
. □
The following remark is in order.
Remark 2. As mention in the Introduction, for , the generalized distance matrix is same as the distance matrix and, for , twice the generalized distance matrix is same as the distance signless Laplacian matrix . Therefore, if in particular we put and , in all the results obtained in Section 3 and Section 4, we obtain the corresponding bounds for the distance energy and the distance signless Laplacian energy .