1. Introduction
Complicated graph structures can often be built from relatively simple graphs via graph-theoretic binary operations such as products. Graph spectrum provides a unique way of characterizing graph structures, sometimes even identifying the entire graph classes. Moreover, using simple graph operations, the spectra of complicated graphs may be constructed from those of small and simple graphs. The interplay between graph spectra (including adjacency, Laplacian, etc.) and various binary graph operations such as corona, edge corona, and disjoint union has been extensively studied in the literature; see e.g., [
1,
2,
3,
4,
5,
6].
In this paper, we consider simple connected graphs [
7]. A graph
G is represented by
, in which the set
represents its vertex set and
is the edge set connecting pairs of distinct vertices. The number
is referred to as the
order of
G and
is the
size of it. A vertix adjacent to a vertex
is called the
neighborhood of
v and is presented by
. The
degree of a vertex
v is the cardinality of its neighborhood and denoted by
or simply
. A
regular graph has the same degree for all vertices. The
distance is the length of a shortest path between two vertices
u and
v. The maximum distance between two vertices is called the
diameter of a graph. The matrix
is called the
distance matrix of
G. As usual,
is the
complement of the graph
G. Moreover, the complete graph
, the complete bipartite graph
, the path
, the cycle
, and the wheel graph
are defined in the conventional way. The sum of the distances from a vertex
v to all other vertices,
, is called the
transmission degree of
v. A
k-
transmission regular graph admits
for any vertex
v. Let
. Then the sequence
is said to be the
transmission degree sequence. The quantity
is referred to as the
second transmission degree of
.
The diagonal matrix
characterizes the vertex transmissions of
G. For a connected graph, M. Aouchiche and P. Hansen [
8,
9] studied the Laplacian and the signless Laplacian for its distance matrix. The
distance Laplacian matrix and the
distance signless Laplacian matrix have attracted great recent research attention due to their usefulness in spectrum theory. Recently, Cui et al. [
10] investigated a convex combination of
and
in the form of
which is called the
generalized distance matrix. Through the study of generalized distance matrix, not only new results can be derived but existing results can be looked into in a new unified point of view.
Let
I be the identity matrix of order
n. The characteristic polynomial of
can be written as
. The
generalized distance eigenvalues of
G are the zeros of
. Noting that
is real and symmetric, we arrange the eigenvalues as
. We call
the
generalized distance spectral radius of
G. The generalized distance spectrum and energy have been recently scoped in [
11,
12].
The rest of the paper is organized as follows. In
Section 2, we study the generalized distance spectrum of join of regular graphs. We will show that the generalized distance spectrum of join of two regular graphs can be obtained from their adjacency spectrum. Again using adjacency eigenvalues, we determine the generalized distance spectrum of join of a regular graph with the union of two different regular graphs. In
Section 3, we use the adjacency matrix eigenvalues and auxiliary matrices to characterize the generalized distance spectrum of the joined union of regular graphs.
2. On the Generalized Distance Spectrum of Join of Graphs
In this section, we study the generalized distance spectrum of join of regular graphs. We will establish new relationship between generalized distance spectrum and adjacency spectrum. As applications, we obtain the generalized distance spectrum of some special graph classes including complete bipartite graph, complete split graph, wheel graph and some derived graphs from a complete graph.
Consider two disjoint vertex sets and with and . For two graphs and , the union is . The join of them is denoted by consisting of and all edges joining each vertex in and each vertex in In other words, the join of them can be obtained by connecting each vertex of to all vertices of
The following gives the generalized distance spectrum of join of two regular graphs in terms of their eigenvalues of adjacency matrices.
Theorem 1. Let be an -regular graph of order , for Let and are the adjacency eigenvalues of and , respectively. The characteristic polynomial of the generalized distance matrix of is given bywhere and . Proof. For
, let
be an
-regular graph of order
. Let
be the join of the graphs
and
. It is clear that
G is graph of diameter 2. Let
be the vertex set of the graph
, then the vertex set of
G is
. For all
, we have
and for all
, we have
. Let us label the vertices of
G, so that the first
vertices are from
. Under this labelling, it can be seen that the generalized distance matrix of
G can be written as
where
,
,
is an all one matrix,
is the identity matrix of order
,
is the adjacency matrix of
and
is the adjacency matrix of the complement
, for
Since
is an
-regular graph, it follows that
, the all ones vector of order
, is an eigenvector corresponding to the eigenvalue
of
and corresponding to the eigenvalue
of
. Let
x be a vector orthogonal to
, satisfying
, then
. Taking
and using
, we have
. This shows that
is an eigenvalue of
corresponding to the eigenvalue
of
. Let
y be a vector orthogonal to
, satisfying
, then
. Taking
and using
, we have
. This shows that
is an eigenvalue of
corresponding to the eigenvalue
of
. The equitable quotient matrix of
is
Since the characteristic polynomial of
M is
and any eigenvalue of
M is an eigenvalue of
[
13], the result follows. □
Let be the complete bipartite graph. It is well-known that . We have the following observation from Theorem 1, which gives the generalized distance spectrum of .
Corollary 1. The generalized distance eigenvalues of consists of the eigenvalue with multiplicity , the eigenvalue with multiplicity and the eigenvalues .
Proof. Similarly as in Theorem 1, this can be proved by taking and , for all . □
Let be the wheel graph of order . It is well known that . Using the fact that the adjacency spectrum of is , we have the following observation from Theorem 1, which gives the generalized distance spectrum of .
Corollary 2. The generalized distance eigenvalues of the wheel graph consists of the eigenvalues and also the eigenvalues .
Proof. Proof follows from Theorem 1, by taking and for . □
The graph of order n is called complete split graph. It is constructed by linking each vertex of a clique of t vertices to each vertex of an independent set of vertices. It is clear that . Using the fact that the adjacency spectrum of is , we have the following observation from Theorem 1, which gives the generalized distance spectrum of .
Corollary 3. The generalized distance eigenvalues of consists of the eigenvalues with multiplicity , the eigenvalue with multiplicity and the eigenvalues, .
Proof. Similarly as in Theorem 1, this can be shown by taking , for and , for . □
In the next result, we work out the relationship between the generalized distance spectrum of the join of regular graphs and their adjacency spectra.
Theorem 2. For let be -regular with order . Let be their adjacency matrices and the adjacency eigenvalues are . We have that the generalized distance spectrum of is eigenvalues for and for and where and three extra eigenvalues defined by the eigenvalues of the following matrixwhere and Proof. Given
. Assume
is
-regular and has
vertices. Let
be the join of the graphs
and
. Obviously,
G has diameter 2. Let
be the vertex set of the graph
, then the vertex set of
G is
. For all
, we have
, for all
, we have
and for all
, we have
. Let us label the vertices of
G, so that the first
vertices are from
, the next
vertices are from
and the next
vertices are from
. Under this labelling, the generalized distance matrix of
G has the form
where
and
for
For a regular graph , the all ones vector of order is an eigenvector corresponding to the eigenvalue . Other eigenvectors are orthogonal to . Therefore, the all ones vector of order is an eigenvector corresponding to the eigenvalue . Other eigenvectors are orthogonal to Suppose that be an eigenvalue of adjacency matrix of and its eigenvector is x satisfying then is an eigenvector of with the eigenvalue Let be any eigenvalues of the adjacency matrix of and with associated eigenvector y and z satisfying , respectively. In a similar way, it can be seen that the vectors and are eigenvectors of with corresponding eigenvalues and respectively.
Hence, we obtained eigenvectors and . They are eigenvectors. It is easy to see that they are orthogonal to and All other three eigenvectors of can be represented by for some
Suppose that
is an eigenvalue of the matrix
with associated eigenvector
. Recall that
and
(
). We obtain:
These equations admit a nontrivial solution only if (
1) has an eigenvalue
. Moreover, any nontrivial solution of the equations is an eigenvector of
associated to
As the remaining three eigenvectors of
are formed like this, it is obvious that any eigenvalue of (
1) is also an eigenvalue of
. □
Consider the graph . We have the following observation from Theorem 2, which gives the generalized distance spectrum of .
Corollary 4. The generalized distance eigenvalues of consists of eigenvalue , with multiplicity , the eigenvalue with multiplicity , the eigenvalue with multiplicity and three more eigenvalues which are the eigenvalues of the matrixwhere . Proof. Proof follows from Theorem 2, by taking , for all and . □
Suppose we have a complete graph of order n. The graph is obtained by removing an edge e from . Taking and , in Corollary 4, we obtain the generalized distance spectrum of the graph given by , where and are the roots of the equation .
3. The Generalized Distance Spectrum of the Joined Union
In this section, we describe the relationship between generalized distance spectrum and the adjacency spectrum of the joined union of regular graphs.
The spectrum of a graph may determine the class of graphs that share the same properties. There have been some different names for the binary graph operation to be introduced below. We will call it joined union following [
4,
6]. This operation is also called generalized composition [
14] or
H-join [
3]. Let
have order
n and
have order
for
. The
joined union is the graph
satisfying:
Clearly, the joined union graph can be constructed by taking the union of and linking any pair of vertices between and if and are neighbors in By this definition, the usual join of and can be viewed as , which is a special joined union graph.
Theorem 3. Suppose G is a graph with diameter at most 2 over . Denote by an -regular graph of order and adjacency eigenvalues where The generalized distance spectrum of the joined union consists of the eigenvalues for and , where and . The remaining n eigenvalues are given by the matrixwhere Proof. Let
G be a graph over
and let
be the vertex set of graph
, for
. Suppose that
is the joined union of the graphs
. By appropriately labelling the vertices of the graph
H, we see that the generalized distance matrix
of the graph
H can be put into the form
where for
is the all-one matrix,
is the adjacency matrix, and
is the identity matrix of order
.
Since
is
-regular, the all-one vector
is an eigenvector of
associated to eigenvalue
. The rest of the eigenvectors turn out to be orthogonal to
We do not require connectivity of
and likewise we do not require
to be a simple eigenvalue. Suppose that
is an eigenvalue of
associated with the eigenvector
satisfying
Note that
X is essentially defined over
and allows a correspondence from
to
. Namely,
(
,
). Given the vector
, where
It can seen that the vector
Y is an eigenvector of
corresponding to the eigenvalue
There exists a total of
mutually orthogonal eigenvectors of
in this manner. They turn out to be orthogonal to the vectors
where
and
This implies that the rest
n eigenvectors of
are spanned by the vectors
which due to the fact that
appear to be linearly independent, suggests that the rest eigenvectors of
are
for some coefficients
Assume that
is an eigenvalue of
associated to an eigenvector
As
(
)
We derive the following equations involving
This set of equations admits a nontrivial solution only if
becomes an eigenvalue of (
2). Moreover, any nontrivial solution of (
3) appears to be an eigenvector of
associated to the eigenvalue
We see that each eigenvalue of (
2) must also be an eigenvalue of
since the rest
n eigenvectors of
are represented in this manner. □
The
lexicographic product of two graphs
G and
H can be constructed in the following way. The vertex set of
is equivalent to the product set
. If
, then
and
are connected, namely, they form an edge in
. We know that
is a special case of joined union
with
(
). When
, it can be seen that
. In view of Theorem 3, the generalized distance spectrum of the joined union
can be written using eigenvalues of
’s as well as those of (
2). The relationship between the eigenvalues of
and the generalized distance spectrum of the joined union
is not explicit though. The following example should shed a light on this relationship. When both
G and
H are regular graphs and
G is a graph of diameter less than or equal to 2, the general distance spectrum of
can be calculated via Theorem 3.
Corollary 5. Suppose that G is s-regular over n vertices with adjacency eigenvalues and diameter less than or equal to 2. Assume that H is r-regular over m vertices with adjacency eigenvalues . Therefore, the generalized distance spectrum of contains for each times) together with the eigenvalues of the matrix , which are and for
It is clear that the complete t-partite graph is a joined union of the graphs , when the parent graph is . That is, . The following observation is a result of Theorem 3 and gives the generalized distance spectrum of, , the complete t-partite graph.
Corollary 6. The generalized distance spectrum of with consists of the eigenvalue , for each times) and the k eigenvalues of the matrixwhere Proof. Proof follows from Theorem 3 by using and the fact that the eigenvalues of are 0 with multiplicity (). □
Example 1. Considering the family of graphs as depicted in Figure 1 and the graph the path of order 3, the generalized distance matrix of the joined union is a block matrix of the formwhere and Since the adjacency spectrums of are and respectively, then from Theorem 3, the generalized distance spectrum of H, consists of the eigenvalues also with the eigenvalues of the matrixTherefore, Note that, as , then the distance spectrum of H isAlso, as , then the distance signless Laplacian spectrum of H is