1. Introduction and Preliminaries
Throughout the paper, let be a connected simple undirected graph with vertex set V and edge set E. The order of a graph is the cardinality of its vertex set, , the size of a graph is the cardinality of its edge set, , and is the adjacency matrix associated with the graph G of order n with eigenvalues for all
The energy of a graph is a concept that comes from theoretical chemistry and is studied in mathematical chemistry to approximate the total
-electron energy of a molecule [
1,
2,
3]. In 1978, Ivan Gutman defined the energy of a graph through the eigenvalues of the adjacency matrix of a graph [
4] as
It should be noted that the interest for this graph invariant goes far beyond chemistry; for example, it is used for minimum and maximum energy for certain families of graphs [
5,
6,
7], on upper and lower bounds for the energy of graphs [
8,
9,
10,
11], and the extension of energy to the Laplacian energy of graphs [
12]. Additionally, in [
13], the authors list various papers published in 2019 on graph energies, and summarize their main collective characteristics, but an interesting motivation is that the authors cite articles that show applications of graph energy outside the field of mathematics. For example, in [
14], a topic related to climate change is addressed; the authors identify the vertices of a graph with the terms “soil”, “climate”, “hydrogeomorphic features”, “biotic features”, and similar, connected by directed or undirected edges, and, in this case, the respective energy “indicates the overall strength of positive and negative feedbacks present”.
In 2008, Liu and Liu defined the Laplacian-energy-like of a graph [
15] as
where
are the Laplacian eigenvalues of
G.
The authors in [
15] have shown that this invariant has similar features to molecular graph energy, see [
4]. In [
16], it was demonstrated that
can be used both in graph discriminating analysis and in correlating studies for modeling a variety of physical and chemical properties and biological activities. This motivates us to extend the results of
by considering other matrices associated with graphs that can be representations of various physical or chemical phenomena.
The distance between the vertices
and
of
G, denoted by
, is equal to the length of (number of edges in) the shortest path that connects
and
. The Harary matrix of graph
G, which is also called the reciprocal distance matrix, and is denoted by
, was defined in 1993, independently in [
17,
18], as a matrix of order
n, given by
Henceforth, we consider for .
The reciprocal distance degree of a vertex
v, denoted by
, is given by
Let be the diagonal Harary degree matrix of order n defined by for .
The Harary index of a graph
G, denoted by
, is defined in [
17,
18] as
In 2018, the authors defined the reciprocal distance Laplacian matrix [
19] as
Since
is a real symmetric matrix, we can write its eigenvalues in decreasing order
We observe that is a positive semi-definite matrix. The following result is fundamental in matrix theory, and will help us to show that the smallest eigenvalue of is simple, i.e., their algebraic multiplicity is one.
Theorem 1 ([
3] Eigenvalue Interlacing Theorem)
. Let A be the symmetric matrix of order n. Let B be a principal sub-matrix of order m, obtained by deleting both i-th row and i-th column of A, for some values of i. Suppose A has eigenvalues and B has eigenvalues . Thenfor . A fundamental spectral result of the Laplacian matrix associated with a graph is that its smallest eigenvalue is zero. The following result shows that for the extension , zero is also an eigenvalue.
Lemma 1. Let G be a connected graph on n vertices and let be the all one vector. Then, is an eigenpair of and 0 is a simple eigenvalue.
Proof. Clearly, each row sum of is 0. Thus, is an eigenpair of . Now, we show that 0 has multiplicity equal to 1.
If , the result is immediate.
If
, then let
be the
matrix obtained from
by eliminating the
i-th row and column. Using Theorem 1, we have that
On the other hand, by construction, we can see that the matrix is strictly diagonal dominant, so is a positive definite matrix. Therefore, . □
In [
20], the authors established bounds for the spectral radius of reciprocal distance Laplacian matrix, and in [
21] gave bounds on the reciprocal distance Laplacian energy and characterized the graphs that attained some of those bounds.
Since is a positive semi-definite matrix, then the spectral radius .
We recall that the notation means that the eigenvalue has an algebraic multiplicity equal to p.
Remark 1. We can see that for the complete graph , where is a matrix with all its entries equal to one and denotes the identity matrix of order n. Then, the -spectrum is .
Theorem 2 ([
19])
. If G is aconnected graph on vertices, then the multiplicity of is with equality if and only if G is the complete graph. Theorem 3 ([
19])
. Let G be a connected graph on n vertices and edges. Consider the connected graph obtained from G by the deletion of an edge. Letandbe the reciprocal distance Laplacian spectra of G and , respectively. Then, for all , An immediate consequence of Theorem 3 is the following result.
Corollary 1. Let G be a connected graph on n vertices. Then At least to the authors’ knowledge, there are several upper bounds, but we only found one lower bound, which is the below theorem
Theorem 4 ([
20])
. Let G be a connected graph on vertices. Then Clearly, when n increases, the bound is not adjusted, so in this work we will propose a more appropriate bound. For this, it is necessary to remember the definition of the Frobenius norm of a matrix.
The Frobenius norm of an
matrix
is
Finally, we give the definition of the equitable quotient matrix and its respective spectral result, which will be a fundamental tool in the development of this work.
Definition 1 ([
22])
. Let X be a complex matrix of order n described in the following block formwhere the blocks are matrices for any and . For , let denote the average row sum of , i.e., is the sum of all entries in divided by the number of rows. Then is called the quotient matrix of X. If for each pair , has a constant row sum, then is called the equitable quotient matrix of X. Theorem 5 ([
23])
. Let be the equitable quotient matrix of X as defined in Definition 1. Then, the spectrum of is contained in the spectrum of X. In this paper, applying the block division matrix technique and using the quotient matrix, new spectral results are obtained from the reciprocal Laplacian matrix. Additionally, we obtain the eigenvalues of certain graphs and in particular obtain eigenvalues of the generalized join product of regular graphs. Finally, we apply these results to obtain extreme graphs, among all the connected graphs of prescribed order in terms of the vertex connectivity, for the energy that we define as Harary Laplacian-like energy, denoted by
as
2. Main Spectral Results
In this section, we give some spectral results about the matrix. In particular, we give a new lower bound for the spectral radius and we characterize the spectrum of certain types of graphs. Our result is more appropriate than the one given by Theorem 4, since for any value of n it is positive and we also characterize when this bound is obtained.
Theorem 6. Let G be a connected graph on vertices. Then The equality holds if and only if .
Proof. Since
G is a connected graph of order
, then, from Lemma 1, for
and
The equality holds if and only if the equality (
1) holds, and this is
Therefore, the multiplicity of is . Using Theorem 2, we conclude that . □
A pair of vertices u and v in G are called twins if they have the same neighborhood, and the same edge weights in the case of a weighted graph. Twin vertices in graphs have proven very useful in the study of the spectra. Motivated by the study of eigenvalues of the adjacency and Laplacian matrices when there are twin vertices in a graph, we study the eigenvalues for the matrix .
If is an edge in G, they are called adjacent twins, and if is not an edge in G, they are called non-adjacent twins.
Theorem 7. Let G be a connected graph of order n and U be a subset of such that U is a set of non-adjacent twins, with . Then is constant for each and is an eigenvalue of with a multiplicity of at least .
Proof. Without loss of generality, we can label the vertices of
U as
. Then
for all
and for all
, in particular
for all
. We note that
for all
. Let
for all
and let
for
where
is the
i-th canonical vector. Then
Since are linearly independent, is an eigenvalue of with a multiplicity of at least . □
Proposition 1. Let be a tree of diameter 3 and order . Then the -eigenvalues are and the eigenvalues of the matrix Proof. Let
u,
v be the central vertices of a tree of diameter 3 (see
Figure 1). Consider the following partition of the graph
Clearly, and are independent sets to such that for any , and for any , where denotes the neighboring vertices of
We observe that for all vertex in we have , analogously for all vertex in , and . Then, using Theorem 7, we have that and are eigenvalues of with multiplicities and , respectively.
Let
and
. Then,
has the following block matrix form
Let
be the equitable quotient matrix of
considering that each block has a constant row sum, an then we can write
Applying Theorem 5, the eigenvalues of are eigenvalues of . Finally, replacing , , and , the matrix given in Theorem is obtained. □
A set of vertices that induces a subgraph with no edges is called an independent set. A bipartite graph is a graph whose vertex set can be partitioned into two disjoint and independent sets and , that is, every edge connects a vertex in to one in . A complete bipartite graph is a special kind of bipartite graph where every vertex of the first set is adjacent to every vertex of the second set. A complete bipartite graph with partitions of size and is denoted by . A star graph is a complete bipartite graph with partitions of size and . The star graph of order n is denoted either by or .
The following proposition gives us the eigenvalues of a complete bipartite graph associated to the reciprocal distance Laplacian matrix.
Proposition 2. Let be a complete bipartite graph on vertices. Then, the spectrum of is Proof. Since and are independent sets to such that for any , and for any , , then for all , we obtain that and for all vertex we have that . Now, using Theorem 7, we have that is an eigenvalue with multiplicity and is an eigenvalue with multiplicity .
From Lemma 1, we have that zero is always an eigenvalue of the matrix . We observe that and are non-zero. Therefore, we have eigenvalues. We obtain the other eigenvalue by applying the fact that the trace of the matrix is equal to the sum of its eigenvalues and the trace of is equal to the sum of its reciprocal distance degrees. □
Corollary 2. Let be the star graph on vertices. Then the spectrum of the reciprocal distance Laplacian matrix of is Theorem 8. Let G be a connected graph of order n and U be a subset of such that U is a set of adjacent twins, with . Then, is constant for each and is an eigenvalue of with multiplicity at least .
Proof. Without loss of generality, we can label the vertices of
U as
. Then
for all
and for all
, and in particular
for all
. Thus,
for all
. Let
for all
and let
for
. Then,
Since are linearly independent, is an eigenvalue of with a multiplicity of at least . □
A clique of a graph is a subset of vertices in which every two vertices are adjacent. A complete split graph, denoted by , is a graph consisting of a clique of a vertices and an independent set of the remaining vertices, such that each vertex in the clique is adjacent to every vertex in the independent set.
Proposition 3. Let be a complete split graph on n vertices. Then Proof. Notice that for the clique of order
a, we have that
Applying Theorem 8, we obtain that n is an -eigenvalue of with a multiplicity of at least .
Analogously, for the independent set of order
, we have that
Applying Theorem 7, we obtain that
is an
-eigenvalue of
with a multiplicity of at least
.
From Lemma 1, we have that zero is always an eigenvalue of the matrix . Since , we have eigenvalues. We obtain the other eigenvalue by applying the fact that the trace of the matrix is equal to the sum of its eigenvalues, and the trace of is equal to the sum of its reciprocal distance degrees. □
Finally, in this section, we study the spectrum for the generalized join graph operation. In particular, we characterize the spectrum of the reciprocal distance Laplacian matrix for the generalized join of regular graphs. We study this operation between graphs since certain graphs can be expressed as a join product of graphs of a much lower order, reducing the process of obtaining their eigenvalues. Furthermore, in the final part of the next section, we use the join product of certain types of graphs to obtain bounds for the Harary Laplacian-energy-like.
Let
and
be two vertex disjoint graphs. The join graph operation of
and
is the graph
such that
and
. This join operation can be generalized as follows [
24]: Let
be a graph of order
s. Let
. Let
be a set of pairwise vertex disjoint graphs. For
, the vertex
is assigned to the graph
. Let
G be the graph obtained from the graphs
and the edges connecting each vertex of
with all the vertices of
if and only if
. That is,
G is the graph with vertex set
and edge set
This graph operation is denoted by
Consider the vertices of G with the labels starting with the vertices of , continuing with the vertices of and finally with the vertices of .
Let
be a connected graph of order
s and for
, let
be a
-degree regular connected graph of order
. Let
be the distance between
. Here, with the above mentioned labeling, we obtain that the reciprocal distance Laplacian matrix of the
-join
has the form
where
with
Notice that the matrices
defined in (
3) are not necessarily the reciprocal distance Laplacian matrices
.
Now, using the results of the partitioned quotient matrix, we will obtain the spectrum of the reciprocal distance Laplacian of the graph , where is a family of a regular graph.
Lemma 2. Let be the diagonal blocks of the matrix defined in (2) such that, for , is a -regular graph with adjacency eigenvalues . Then, the spectrum of iswith . Proof. Since
has constant row sum equal to
where
, and there is
l such that
with
Let
be the set of eigenvectors corresponding to the adjacency eigenvalues of regular graph
,
respectively. We observe that, for
,
and
. Then
Therefore, for
the spectrum of
is
with
. □
The following result allows us to know the spectrum of new families of regular graphs obtained using the generalized graph join operation.
Theorem 9. Let be a connected graph of order s and where, for each , is a regular graph. Then, the spectrum of iswhere and is the matrix Proof. Let
be a
-regular graph such that
is a set of eigenvectors corresponding to the adjacency eigenvalues
. Then, using Lemma 2 we have that, for
,
Therefore, for
and for
,
The remaining eigenvalues will be obtained from the quotient matrix associated with the matrix
given in Equation (
2). The matrix
is a matrix partitioned into blocks such that each block has a constant row sum. Then, the equitable quotient matrix of
is
So, applying Theorem 5, we obtain that
□
Corollary 3. Let be a connected graph of order s and where, for each , is a regular graph. Then, Proof. From Equation (
2), we can see that the eigenvalues of the matrices
interlace the eigenvalues of
. Then, from Theorem 9, the largest and smallest eigenvalues of
are the largest and smallest eigenvalues of
, respectively. □
Example 1. For , , and , the graph is given in the following Figure 2. In this example, we have that , , and Thus, , , where denotes that λ is an eigenvalue with multiplicity t. Moreoverand . Applying Theorem 9, we obtain A wheel graph on vertices, denoted by , is a graph formed by connecting a single vertex to all vertices of a cycle of order . We observe that .
A generalized wheel graph on
n vertices, denoted by
, is a graph consisting of an independent set of order
a and a cycle of the remaining
vertices, such that each vertex in the cycle is adjacent to every vertex in the independent set. We can write this graph as
The
is not a connected graph, and since we define the join product for connected graphs, so that the distance between the components is not indeterminate, then we will use the following notation
Proposition 4. Let be a generalized wheel graph on n vertices. Then the eigenvalues of are and the eigenvalues of the form for .
Proof. In the generalized join operation
we associate the cycle to the central vertex of the star and each vertex
is associated to the pending vertices of the star. Considering the labeling in this same order, we have that the expressions given in (
2) and (
3), for the graph
, are
where
and
are matrices of order
. Now, expression (
4) for
becomes
Since the adjacency eigenvalues of the cycle of order p have the form , then, from Theorem 9, for , we have that are eigenvalues of .
Now, the matrix
given in Theorem 9 has the form
We recall that
denotes the
i-th canonical vector. Then, for
Thus,
is an eigenvalue of
with multiplicity
. From Theorem 9, we have that
is an eigenvalue of
with multiplicity
. From Corollary 3, we have that the largest and smallest eigenvalues of
are the largest and smallest eigenvalues of
matrix given in (
5). Then, we obtain the spectral radius applying the fact that the trace of the matrix is equal to the sum of its eigenvalues. This fact, applied to the matrix
, gives us that
. □
An immediate consequence is the following result.
Corollary 4. Let be a wheel graph on n vertices. Then, the eigenvalues of are and the eigenvalues of the form for .
Example 2. Consider the wheel graph (see Figure 3). The spectrum . In fact, verifying the expressions given in Corollary 4, we see thatand, for , so we apply the expression Example 3. Consider the following generalized wheel graph with , (see Figure 3). The spectrum Verifying the expressions given in Proposition 4, we obtain For applying the expressionwe have 3. Bounds for Harary Laplacian-Energy-like
In this section, we obtain upper and lower bounds of Harary Laplacian-energy-like of a graph, in terms of the other invariants of the graph.
Remark 2. If is the connected graph obtained from G by the deletion of an edge, then from definition of the reciprocal distance Laplacian matrix, we obtain that .
An immediate consequence due to the Theorem 3 is the following result.
Corollary 5. If G and are connected graphs such that is obtained from G by the deletion of an edge, then From Corollary 5 and Remark 1, we obtain the following result.
Corollary 6. Among the all connected graphs on n vertices and edges, the complete graph has the largest Harary Laplacian-energy-like. Thuswith equality if and only if . In the following Theorem, we obtain an upper bound of Harary Laplacian-energy-like in terms of the number of vertices and the Harary index.
Theorem 10. Let G be a connected graph of order and Harary index . Thenwith equality if and only if . Proof. Let
G be a connected graph of order
. We have
From Theorem 3 and Remark 1, we obtain
Replacing Equation (
6), we obtain
We observe that the equality occurs if and only if and , for all . Applying Theorem 2, we obtain that . □
An upper bound of Harary Laplacian-energy-like in terms of the number of vertices, Harary index and the Frobenius norm of reciprocal distance Laplacian matrix of graph G are obtained.
Theorem 11. Let G be a graph of order . Then The equality holds if and only if .
Proof. We observe that
. Using the Cauchy–Schwartz’s inequality, we obtain
Let be a real function. We recall that is a strictly decreasing function in the interval .
We observe that
and
Then, using Theorem 6, we have
Now, we observe that the equality holds whenever the equality (
7) and the bound given in Theorem 6 hold. In the first case, the equality occurs if
and thus the multiplicity of
is
and, using Theorem 2, we conclude that
. In the second case, the equality in Theorem 6 occurs if
. Reciprocally, if
with
, then
□
The following example shows a comparison of the bounds obtained with the real values considering various known graphs.
Example 4. We consider the graphs , given by Figure 4. Let be the Petersen graph and let , and be the star, path and cycle on seven vertices, respectively. Using four decimal places, in Table 1 we show the upper bounds obtained for Harary Laplacian-energy-like for the given graphs. Lemma 3 ([
25])
. Let m and n be natural numbers such that . Let be positive real numbers. Then, Theorem 12. Let G be a graph of order . Then Proof. Applying Lemma 3 to the definition of
, we obtain
Example 5. The Table 2 shows, to four decimal places, the lower bound obtained for Harary Laplacian-energy-like for the graphs given in Example 4. Now, to begin the end of this section, we recall that the vertex connectivity of a graph G, denoted by , is the minimum number of vertices of G, the deletion of which disconnects G. Clearly, . Applying the results of the previous section, we find upper bounds on Harary Laplacian-energy-like among all the connected graphs of prescribed order in terms of the vertex connectivity.
Let be the family of connected graphs G of order n such that .
For
, let
be a
-regular graph of order
. Then
is a graph of order
. Observe that
and
Labeling the vertices of
starting with the vertices of
, continuing with the vertices of
and finishing with the vertices of
, and using the results obtained in the previous subsection, the reciprocal distance Laplacian matrix
becomes
where for
and
The eigenvalues of
,
and
associated to
and
, respectively, are
Using these observations, the following result is due to applying Theorem 9.
Proposition 5. If and, for , is a -regular graph thenwhere are as in (8), are as in (9) and Let
n and
k be positive integers, with
and consider the graph
where, without loss of generality, we assume
. The following result is obtained by applying Proposition 5 to
.
Proposition 6. Let such that . Then Proof. We observe that for the graph
, the matrices
,
and
in (
8) are
respectively, and the matrix
in (
10) becomes
Then,
and the spectrum of
is
The result is obtained. □
If, denotes the empty graph, i.e., the graph without edges and without vertices, then we have the following results.
Lemma 4. Let for . Then Proof. From Proposition 6, we have
We observe that
for
and
f is a strictly decreasing function in the interval
Thus, the result is obtained. □
Theorem 13. If , then Additionally, the equality in (11) holds if and only is . Proof. Let
. We first consider
. From Corollary 6,
with equality if and only if
. Furthermore
Then, the result is true for . Now, let and let such that is a maximum.
Let
such that
is a disconnected graph and
. We denote by
the
r connected components of
. Clearly
. Suppose that
, then we can construct a new graph
where
e is an edge connecting a vertex in
with a vertex in
. We can see that
. Using Corollary 5, we have
It is a contradiction because
G is the graph with maximum
. Therefore
, that is,
.
By definition . Now, we claim that .
Suppose
. Since
, we may construct a graph
where
e is an edge joining a vertex
with a vertex
. We see that
is a connected graph and the deletion of the vertex
u disconnected it, and then
. Using Corollary 5,
, which is also a contradiction. Hence,
and
. Let
. Then
. Through the repeated application of Corollary 5, we can conclude that
for some
. We have proved
for all
. From Lemma 4, we have that
. Since
, for
, the equality holds if and only if
. □
Example 6. For and , the graphs with minimum and maximum Harary Laplacian-energy-like are and , respectively (see Figure 5). In fact, using four decimal places,