1. Introduction
The study of spectrum of commuting graphs of a given group or a ring attracted a large amount of attention in recent years. The concept of commuting graph has been defined in [
1]. Let
G be a group and
X be a nonempty subset of
G. The commuting graph
has the vertex set
X where
are adjacent whenever
in
G. In the case that
, we denote
for short. Various aspects of commuting graphs of finite groups can be found in [
2] and its references.
In this paper, we only consider finite simple graphs on vertex set
, i.e., graphs without multiple-edges and loops on a finite vertex set. We use the following standard notation; see, e.g. [
3,
4]. We associate
with a symmetric real
matrix
, called
adjacency matrix, where
if the vertex
is adjacent to
, and 0 otherwise. The characteristic polynomial of
is
. These eigenvalues with their multiplicity compose the spectrum of graph
. Recall that the spectral radius of a matrix
A, denoted by
, is defined by
.
Let denote the diagonal matrix of degrees of a graph . The Laplacian matrix of is the matrix , and the Laplacian spectrum of is a multiset given by L-spec where are the Laplacian eigenvalues of with multiplicities
The
energy of the graph
is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix, that is,
This concept was introduced in 1978 by Gutman [
5]. Clearly, the energy of complete graph
of order
n is
A graph
is said to be
hyperenergetic if
and
nonhyperenergetic if
. It is known [
6] (Theorem 5.24) that for almost all graphs
,
, which means that almost all graphs are hyperenergetic. Therefore, the problem of finding nonhyperenergetic graphs has been extremely meaningful, see [
7]. Additionally, the
Laplacian energy, denoted by
is defined as
where
m is the number of edges in the graph
Both the energy and the Laplacian energy of graphs play an important role in solving many physical and chemical problems. In theoretical chemistry, the energy of a given molecular (conjugated hydrocarbons) graph approximately describes the total
electron energy of the molecule represented by that graph, see the comprehensive survey [
6,
8] for more details.
Let be a simple graph and u be a vertex. The neighbor set of u in , denoted by , is the set of all vertices adjacent to u. The degree of u is The minimum degree of is denoted by . A spanning tree of is a connected subgraph of on all vertices with edges, which is in fact a tree on . The number of spanning trees in a graph is denoted by .
A simple graph is said to be planar if it can be drawn on the plane with no crossing edges. The vertex-connectivity of is denoted by , which is the minimum number of vertices whose deletion will result in a disconnected graph or trivial graph. A set is called a disconnecting set if its deletion will increase the number of components of , and S is called a minimal disconnecting set if there is no proper subset of S that disconnects . For two vertex sets U and V, we denote by the set of edges having one end-point in U and the other in
Our focus in this paper lies on the semidihedral group
for
. Moreover, all the
elements of
could be given as
From Lemma 10 of [
9], we have
for odd
n, and
for even
Remark 1. If n is odd, then is the center of semidihedral group. Now, we suppose that , where and are the subsets of It is easy to check that for all However, when and If n is even, then is the center of semidihedral group. Similarly, we suppose that , where and are the subsets of It is also easy to find that for all However, if and
In the existing literature, there are still many gaps in finding characteristic polynomials, spectral radius, graph energy, Laplacian spectrum, vertex-connectivity, planarity, and Hamiltonian graphs of commuting graphs of finite semidihedral groups . We will address some of these problems in this paper. When n has a large number of factors, it is not easy to obtain explicit expression for the eigenvalues of the graph , so that we resort to bounding the spectral radius in such cases. Moreover, we prove that the commuting graph is not hyperenergetic when odd or even . In addition, we obtain the characteristic polynomial of the Laplacian matrix of , and we also derive the Laplacian spectrum of Moreover, we calculate the Laplacian energy and the number of spanning trees of Finally, we discuss vertex connectivity, planarity, and minimum disconnecting sets of commuting graphs of the semidihedral group and prove that these commuting graphs are not Hamiltonian.
3. Laplacian Spectral Properties of Commuting Graphs
In this section, we are going to find the Laplacian spectrum of commuting graphs on semidihedral groups, and then we compute the Laplacian energy and the number of spanning trees.
Firstly, we obtain the Laplacian matrix of
when
n is odd. Now we have
For notational convenience, we define
and
where
and
F are of order
. And then we denote
then the Laplacian matrix of
is given as
Theorem 3. For odd the characteristic polynomial of the Laplacian matrix of is Proof. By Theorem 2.2 of [
10], we get
Upon calculation, we get from Theorem 2.5 of [
10] that
where
and
. Hence,
Applying a series of row and column operations, that is, for
, subtracting the
i-th row from the
-th row, and subtracting the
i-th column from the
-th column. Then
where
and
For notational convenience, we set
We now are ready to compute the values of
and
. Note that
Thus,
And then we have
Combining
and
, it follows that
which together with
leads to the required result. □
As byproducts of Theorem 3, we obtain the following corollaries.
Corollary 5. For odd the Laplacian spectrum of is Corollary 6. For odd the Laplacian energy of is Corollary 7. For odd , the number of spanning trees of is Proof. The proof is straightforward by using Corollary 5 and Proposition 1.3.4 of [
12]. □
Finally, we are ready to compute the Laplacian spectrum of commuting graphs of
when
n is even, then we have
. For notational convenience, we first define
Here,
and
Z are all of order
. We then denote
and
then
Theorem 4. For even the characteristic polynomial of the Laplacian matrix of is Proof. By Theorem 2.2 of [
10], we have
Upon computations, we get
, and Theorem 2.4 of [
10] yields
Then
where
and
It is not hard to find that
and
Therefore, we conclude that
This completes the proof. □
From Theorem 4, one could get the following corollaries similarly.
Corollary 8. For even the Laplacian spectrum of is Corollary 9. For even the Laplacian energy of is Corollary 10. For even , the number of spanning trees of is 4. Connectivity and Planarity of Commuting Graphs
In this section, we show that these commuting graphs are not Hamiltonian. Furthermore, we discuss vertex connectivity, the planarity, and minimum disconnecting set of commuting graphs of semidihedral groups.
Theorem 5. Suppose is a semidihedral group and is the commuting graph on , for . Then
For odd
For even
is not Hamiltonian for both odd and even
Proof. (1) For an odd integer n, since we know that the center of is adjacent to all other vertices of , so using Remark 1, we have the graph becomes disconnected into components by deletion of . Therefore,
(2) For an even integer since we know that the center of is adjacent to all other vertices of , so using Remark 1, the graph becomes disconnected into components by deletion of . Therefore,
(3) For an odd integer
we have
and the components of
are
Also, for an even integer
n, we have
and the components of
are
Therefore, by Proposition 7.2.3 of [
13], we have
is not Hamiltonian. □
Theorem 6. Let be the commuting graph of the semidihedral group for . Then is nonplanar.
Proof. Note that is cyclic subgroup of order . It suffices to show that the commuting graph is nonplanar if and only if the subgraph induced by is nonplanar. Since is cyclic, is nonplanar for . □
Theorem 7. For odd we have . Moreover, for every we have , , , , , , and , , , are minimum disconnecting sets of
Proof. Suppose so Now let the only vertices adjacent to v in are elements of and so . Hence, we conclude that .
It is clear from the structure of when n is odd, where . Since we have is the minimum disconnecting set of Similarly, we have , , , , and , , , are minimum disconnecting sets of □
Theorem 8. For even we have . Moreover, for every we have and are minimum disconnecting sets of
Proof. Similar to the proof of Theorem 7. □