1. Introduction
This paper considers only finite undirected simple graphs. Let G be a graph with order n, its the adjacency matrix is defined as follows:
where i~j denotes that the vertex i and the vertex j are adjacent. Obviously,
is a real symmetric matrix in which all diagonal elements are 0 and all other elements are 0 or 1, its eigenvalues are all real numbers. The n eigenvalues of
are said to be the eigenvalues of the graph G and to form the spectrum of this graph. Let
be the distinct eigenvalues of the graph G, and their multiplicities are
, respectively, then
. Throughout this paper, the spectrum of the graph G is often labeled as
. Let
be an eigenvalue of the graph G, a subspace of the vector space
in the complex field generated by all the vectors x that satisfy
is called the eigensubspace of the eigenvalue
, and we denote it as
. Its dimension is denoted by
, which is equal to the multiplicity of the eigenvalue
.
Since the adjacency matrix
is a real symmetric matrix, there exists a unitary matrix U such that
, where
is a diagonal marix, and the columns of the matrix U are the eigenvetors corresponding to the eigenvalue, and they form a standard orthonormal basis of the vector space
,
denotes the conjugate transpose of the matrix U. If this basis is constructed by stringing together the orthonormal basis of the eigenspaces of A, then
, where
are the distinct eigenvalues of A and each
has block diagonal form
. Then A has the spectral decomposition [1] [2]
(1)
where
. For fixed i, if
has
as a unit orthonormal basis, then
(2)
and
represents the orthonormal projection of
on
. Moreover, it satisfies
,
,
.
The module
of the orthonormal projection of the vector
into the eigensubspace
is exactly equal to the cosine of the (acute) angle between the vector
and the eigensubspace
, in [2], which is defined as the angle of the graph, and we denote it as
. The
matrix
formed by the angles is called the angle matrix of the graph G. The module
of the orthonormal projection of the unit vector
into the eigensubspace
is exactly equal to the cosine of the (acute) angle between the vector j and the eigensubspace
, in [1], which is defined as the main angle of the graph, and we denote it as
. The vector
formed by the main angles is called the main angle vector of the graph G, where
.
For positive integers
,
,
and
denote the complete graph on n vertices, the n-cycle and the complete bipartite graph on
vertices, respectively.
denotes the cube graph on 8 vertices (as shown in Figure 1), and P denotes the Petersen graph (as shown in Figure 2). There are many studies on the eigenvalues of the graph in [1] [2] [3] [4] [5], since the angles and
Figure 1. The eigenvectors corresponding to the eigenvalues 1, −1 and −3 of Q8. (a) The weight eigenvectors corresponding to 1; (b) The weight eigenvectors corresponding to −1; (c) The weight eigenvectors corresponding to −3.
Figure 2. The eigenvectors corresponding to the eigenvalue 1 of P.
main angles are difficult to determine, there is a few study on them. Actually, they are all important parameters on the graph, and they can be combined with the eigenvalues of the graph to determine the degree sequence of the graph, the number of triangles, quadrilaterals and pentagons on the graph, and the characteristic polynomials of the complement graph [1] [2]. In this paper, we determine the angles and main angles of the complete graph, the cube graph, the petersen graph, the cycle and the complete bipartite graph. Undefined concepts and notations will follow [1].
2. Some Lemmas
Lemma 2.1 [1] Any graph is determined by its eigenvalues and a basis of the corresponding eigenvectors.
Let G be a graph on n vertices, and its vertices are labeled as
. We assign a weight
to each vertice on the graph, and we get a vector
.
Lemma 2.2 [1] The nonzero vector
is an eigenvector of the graph G corresponding to the eigenvalue
if and only if
holds, where
is the sum of the weights
of all vertices j adjacent to the vertice i.
Remark Lemma 2.2 shows that the nonzero vector
is an eigenvector of the graph G corresponding to eigenvalue
if and only if the
times of the weight
of the
th vertice is exactly equal to the sum of the weight of each vertice adjacent to it.
Lemma 2.3 [1] The angles
of a graph satisfy the equalities
Lemma 2.4 [1] The main angles
of a graph satisfy the equality
3. The Primary Outcomes
Theorem 3.1 The angle matrix of the complete graph
is
the main angle vector is
.
Proof. By [5], we know that the eigenvalues of
are
and −1, the multiplicities of them are 1 and
, respectively. So the spectrum of
is
the eigenvector corresponding to the eigenvalue
is the all −1 vector
, and we unify it into
. So the matrix
, by the definition of the angle,
. By Lemma 2.3, we can easily get the second row of the angle matrix
. By the definition of the main angle, and
,
. By Lemma 2.4, we can easily get the main angle vector
.
Theorem 3.2 The angle matrix of the cube graph
is
the main angle vector is
.
Proof. By [4], we know that the eigenvalues of
are 3, 1, −1 and −3, the multiplicities of them are 1, 3, 3 and 1, respectively. So the spectrum of
is
the eigenvector corresponding to the eigenvalue 3 is the all −1 vector
, and we unify it into
. So the matrix
, by the definition of the angle,
. By the definition of the main angle,
. So all angles corresponding to the eigenvalue 3 are
, the main angle is 1.
By Lemma 2.1 and Lemma 2.2, we can know that the orthonormal eigenvectors corresponding to the eigenvalue 1 are (as shown in Figure 1(a))
and we unify them into
, then the orthonormal projection matrix
by the definition of the angle,
in the same way,
Since
are orthonormal to the all −1 vector, then
So all angles corresponding to the eigenvalue 1 are
, the main angle is 0.
By Lemma 2.1 and Lemma 2.2, we can know that the orthonormal eigenvectors corresponding to the eigenvalue −1 are (as shown in Figure 1(b))
and we unify them into
, then the orthonormal projection matrix
then
Since
are orthonormal to the all −1 vector, then
So all angles corresponding to the eigenvalue −1 are
, the main angle is 0.
By Lemma 2.1 and Lemma 2.2, we can know that the orthonormal eigenvector corresponding to the eigenvalue −3 is
(as shown in
Figure 1(c)), and we unify them into
. So the orthonormal projection matrix
, then
So all angles corresponding to the eigenvalue −3 are
, the main angle is 0.
Theorem 3.3 The angle matrix of the Petersen graph P is
the main angle vector is
.
Proof. By [4], we know that the eigenvalues of P are 3, 1 and −2, the multiplicities of them are 1, 5 and 4, respectively. So the spectrum of P is
the eigenvector corresponding to the eigenvalue 3 is the all −1 vector
, and we unify it into
. So the matrix
, then
So all angles corresponding to the eigenvalue 3 are
, the main angle is 1.
By Lemma 2.1 and Lemma 2.2, we can know that the eigenvectors corresponding to the eigenvalue 1 are (as shown in Figure 2)
and we unify and orthogonality them into
Then the orthonormal projection matrix
,
in the same way, we have
Since
are orthonormal to the all −1 vector j, then
So all angles corresponding to the eigenvalue 1 are
, the main angle is 0.
By Lemma 2.1 and Lemma 2.2, we can easily know that all angles corresponding to the eigenvalue −2 are
, the main angle is 0.
Theorem 3.4 The angle matrix of the n-cycle
is
the main angle vector is
.
Proof. By [1], we know that the eigenvalues of
are
, the multiplicities of them are all 1. So the spectrum of
is
when
, the eigenvector corresponding to the eigenvalue 2 is the all −1 vector
, and we unify it into
. So the orthonormal projection matrix
, then
So all angles corresponding to the eigenvalue 2 are
, the main angle is 1.
By Lemma 2.1 and Lemma 2.2, we can know that the eigenvectors corresponding to the eigenvalue
are
where
, then the orthonormal projection matrix
, for example, when
, we have the orthonormal projection matrix
corresponding to the eigenvalue
,
Since the eigenvectors corresponding to distinct eigenvalues are orthonormal, then
So all angles corresponding to the eigenvalue
are
, the main angle is 0.
In the same way, all angles corresponding to the eigenvalue
are
, the main angle is 0.
Theorem 3.5 The angle matrix of the complete bipartite graph
is
the main angle vector is
.
Proof. By [2], we know that the eigenvalues of
are
, 0 and
, the multiplicities of them are 1,
and 1, respectively. So the spectrum of
is
The vertices of complete bipartite graph
can be divided into two color
classes X and Y,
,
. If we assign a weight
to each vertex of X and a weight
to each vertex of Y, by Lemma 2.2, we can know that the vector
is a unit eigenvector corresponding to the eigenvalue
, so the orthonormal projection matrix corresponding to the eigenvalue
is
, then
So all angles corresponding to the eigenvalue
are m
s and n
s, the main angle is
.
If we assign a weight
to each vertex of X and a weight
to each vertex of Y, by Lemma 2.2, we can know that the vector
is a unit eigenvector corresponding
to the eigenvalue
, so the orthonormal projection matrix corresponding to the eigenvalue
is
, then
So all angles corresponding to the eigenvalue
are m
s and n
s, the main angle is
.
By Lemma 2.3, all angles corresponding to the eigenvalue 0 are m
s and n
s. By Lemma 2.4, the main angle corresponding to the eigenvalue 0 is
.
4. Conclusion
In this paper, the angles and main angles of the complete graph, the cube graph, the Petersen graph, the cycle and the complete bipartite graph are determined.
Acknowledgements
This work is supported by National Natural Science Foundation of China (1561056, 11661066), Natural Science Foundation of Qinghai Provence (2016-ZJ-914).