Properties of Commuting Graphs over Semidihedral Groups

: This paper considers commuting graphs over the semidihedral group SD 8 n . We compute their eigenvalues and obtain that these commuting graphs are not hyperenergetic for odd n ≥ 15 or even n ≥ 2. We further compute the Laplacian spectrum, the Laplacian energy and the number of spanning trees of the commuting graphs over SD 8 n . We also discuss vertex connectivity, planarity, and minimum disconnecting sets of these graphs and prove that these commuting graphs are not Hamiltonian.


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 C(G, X) has the vertex set X where x 1 , x 2 ∈ X are adjacent whenever x 1 x 2 = x 2 x 1 in G. In the case that X = G, we denote Γ(G) = C(G, G) 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 {v 1 , v 2 , . . . , v n }, 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 Γ with a symmetric real n × n matrix A(Γ) = (a ij ) n i,j=1 , called adjacency matrix, where a ij = 1 if the vertex v i is adjacent to v j , and 0 otherwise. The characteristic polynomial of Γ is p(λ) = det(λI − A(Γ)). These eigenvalues with their multiplicity compose the spectrum of graph Γ. Recall that the spectral radius of a matrix A, denoted by ρ(A), is defined by ρ(A) = max{|λ i (A)| : i = 1, 2, . . . , n}.
The energy of the graph Γ is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix, that is, E(Γ) = ∑ n k=1 |λ k (Γ)|. This concept was introduced in 1978 by Gutman [5]. Clearly, the energy of complete graph K n of order n is 2(n − 1). A graph Γ is said to be hyperenergetic if E(Γ) > E(K n ) = 2(n − 1) and nonhyperenergetic if E(Γ) ≤ 2(n − 1). It is known [6](Theorem 5.24) that for almost all graphs Γ, E(Γ) > (1/4 + o(1))n 3/2 , 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 LE(Γ), is defined as LE(Γ) = ∑ l i=1 |µ i − 2m n |, where m is the number of edges in the graph Γ. 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 π−electron energy of the molecule represented by that graph, see the comprehensive survey [6,8] for more details.
Let Γ be a simple graph and u be a vertex. The neighbor set of u in Γ, denoted by N(u), is the set of all vertices adjacent to u. The degree of u is d(u) = |N(u)|. The minimum degree of Γ is denoted by σ(Γ). A spanning tree of Γ is a connected subgraph of Γ on all vertices with |V(Γ)| − 1 edges, which is in fact a tree on V(Γ). The number of spanning trees in a graph Γ is denoted by st(Γ).
A simple graph is said to be planar if it can be drawn on the plane with no crossing edges. The vertex-connectivity of Γ is denoted by κ(Γ), which is the minimum number of vertices whose deletion will result in a disconnected graph or trivial graph. A set S ⊂ E(Γ) is called a disconnecting set if its deletion will increase the number of components of Γ, and S is called a minimal disconnecting set if there is no proper subset of S that disconnects Γ. For two vertex sets U and V, we denote E[U, V] by the set of edges having one end-point in U and the other in V.
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 SD 8n . 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 Γ(SD 8n ), so that we resort to bounding the spectral radius in such cases. Moreover, we prove that the commuting graph Γ(SD 8n ) is not hyperenergetic when odd n ≥ 15 or even n ≥ 2. In addition, we obtain the characteristic polynomial of the Laplacian matrix of Γ(SD 8n ), and we also derive the Laplacian spectrum of Γ(SD 8n ). Moreover, we calculate the Laplacian energy and the number of spanning trees of Γ(SD 8n ). Finally, we discuss vertex connectivity, planarity, and minimum disconnecting sets of commuting graphs of the semidihedral group SD 8n and prove that these commuting graphs are not Hamiltonian.

Spectral Properties of Commuting Graphs
In the present section, we are going to find the characteristic polynomials of commuting graphs of semidihedral groups and discuss their spectral radius. Additionally, we also obtain the lower and upper bounds on the energy of the commute graph Γ(SD 8n ), and prove that Γ(SD 8n ) is not hyperenergetic when odd n ≥ 15 or even n ≥ 2.
Firstly, we can obtain the adjacency matrix of Γ(SD 8n ) in the following. When n is odd, then Γ(SD 8n ) = K 4 ∨ (nK 4 ∪ K 4n−4 ), to derive the adjacency matrix of a commuting graph Γ(SD 8n ), we first put the elements of Z(SD 8n ), then Φ 3 and finally Φ 2 respectively, and get the following matrix is where A(Γ(C 4n )), H 4n and F 4n are of order 4n. When n is even, then we have Here, A(Γ(C 4n )), Y 4n and Z 4n are all of order 4n.
Proof. By Theorem 2.2 of [10], we can get Setting e = (1, 1, 1, 1) T . By directed computation, Theorem 2.5 of [10] gives Applying a series of row and column operations, that is, for i = 2, 3, 4, subtracting the i-th row from the (i − 1)-th row, and subtracting the i-th column from the (i − 1)-th column yields to For notational convenience, we denote Next, we are going to compute the values of Λ 1 , Λ 2 and Λ 3 . Since Therefore And then After calculations, we can get which together with |λI 4n − F 4n | = (λ − 3) n (λ + 1) 3n yields the desired result.

Theorem 2.
For even n ≥ 2, the characteristic polynomial of Γ(SD 8n ) is Proof. By Theorem 2.2 of [10], we get After some computations, we have |λI 4n − Z 4n | = (λ 2 − 1) 2n , and the Theorem 2.4 of [10] gives Thus, we get By subtracting the second row from the first row and subtracting the second column from the first column, we get where Directed calculation gives and We now conclude that Hence, the required result immediately follows. Corollary 3. For even n ≥ 2, the energy of Γ(SD 8n ) satisfies the following, Consequently, the commuting graph Γ(SD 8n ) with even n ≥ 2 is not hyperenergetic.

Corollary 4.
For even n ≥ 2, the spectral radius of Γ(SD 8n ) satisfies the following, Proof. Similar to the proof of Theorem 2.

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 Γ(SD 8n ) when n is odd. Now we have Γ(SD 8n ) = K 4 ∨ (nK 4 ∪ K 4n−4 ). For notational convenience, we define where B, C and F are of order 4n. And then we denote then the Laplacian matrix of Γ(SD 8n ) is given as Theorem 3. For odd n ≥ 2, the characteristic polynomial of the Laplacian matrix of Γ(SD 8n ) is Proof. By Theorem 2.2 of [10], we get Upon calculation, we get from Theorem 2.5 of [10] that Applying a series of row and column operations, that is, for i = 2, 3, 4, subtracting the i-th row from the (i − 1)-th row, and subtracting the i-th column from the (i − 1)-th column. Then For notational convenience, we set We now are ready to compute the values of ∆ 1 , ∆ 2 and ∆ 3 . Note that Thus, And then we have Combining ∆ 1 and ∆ 2 , it follows that which together with |(λ − 7)I 4n + F| = (λ − 4) n (λ − 8) 3n leads to the required result.
As byproducts of Theorem 3, we obtain the following corollaries. 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 SD 8n when n is even, then we have Γ(SD 8n ) = K 2 ∨ (2nK 2 ∪ K 4n−2 ). For notational convenience, we first define Here, X, Y and Z are all of order 4n. We then denote V = 3I 4n − Z and Theorem 4. For even n ≥ 2, the characteristic polynomial of the Laplacian matrix of Γ(SD 8n ) is Proof. By Theorem 2.2 of [10], we have Upon computations, we get |λI 4n − V| = ((λ − 3) 2 − 1) 2n , and Theorem 2.4 of [10] yields 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.

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 SD 8n is a semidihedral group and Γ(SD 8n ) is the commuting graph on SD 8n , for n ≥ 2. Then 1. For odd n, κ(Γ(SD 8n )) = 4; 2. For even n, κ(Γ(SD 8n )) = 2; 3. Γ(SD 8n ) is not Hamiltonian for both odd and even n.

Proof.
(1) For an odd integer n, since we know that the center Z(SD 8n ) = {e, a n , a 2n , a 3n } of SD 8n is adjacent to all other vertices of Γ(SD 8n ), so using Remark 1, we have the graph becomes disconnected into n + 1 components by deletion of Z(SD 8n ). Therefore, κ(Γ(SD 8n )) = 4.
Proof. Note that a is cyclic subgroup of order 4n. It suffices to show that the commuting graph Γ(SD 8n ) is nonplanar if and only if the subgraph induced by a is nonplanar. Since a is cyclic, Γ(SD 8n ) is nonplanar for n ≥ 2.