On some characterizations of strong power graphs of finite groups

Let $ G $ be a finite group of order $ n$. The strong power graph $\mathcal{P}_s(G) $ of $G$ is the undirected graph whose vertices are the elements of $G$ such that two distinct vertices $a$ and $b$ are adjacent if $a^{{m}_1}$=$b^{{m}_2}$ for some positive integers ${m}_1 ,{m}_2<n$. In this article we classify all groups $G$ for which $\mathcal{P}_s(G)$ is line graph and Caley graph. Spectrum and permanent of the Laplacian matrix of the strong power graph $\mathcal{P}_s(G)$ are found for any finite group $G$.


Introduction
Let G be a group of n elements. The strong power graph P s (G) of G is a simple undirected graph whose vertices consists of the elements of G and two distinct vertices a and b are adjacent in P s (G) if a m 1 =b m 2 for some positive integers m 1 , m 2 < n. Thus a finite group G is noncyclic if and only if P s (G) of is complete. The idea of a strong power graph was introduced by Singh and Manilal [16] as a generalization of power graphs of a finite group. Now research on power graphs associated with a finite group has already gained a momentum [4], [5], [12].
Here we study Laplacian spectrum of the strong power graph P s (G) of any finite group G. For any graph Γ let A(Γ) be the adjacency matrix and D(Γ) be the diagonal matrix of vertex degrees.
Then the Laplacian matrix of Γ is defined as L(Γ)=D(Γ) − A(Γ). Clearly L(Γ) is a real symmetric matrix and it is well known that L(Γ) is a positive semidefinite matrix with 0 as the smallest eigen * Corresponding author value. Thus we can assume that the Laplacian eigen values are λ 1 ≥λ 2 ≥λ 3 ≥· · · ≥λ n =0. Among all Laplacian eigen values of a graph, one of the most popular is the second smallest, called by Fiedler [6], the algebraic connectivity of a graph. It is a good parameter to measure, how well a graph is connected. For example, a graph is connected if and only if its algebraic connectivity is non-zero.
The Laplacian matrix of a graph and its eigen values known as Laplacian spectrum are used in different area in mathematics, viz. discrete mathematics, combinatorial optimizations, etc; and to interpret several physical and chemical problems. According to Mohar [13] the Laplacian eigen values are more intuitive and much more important than the eigen values of the adjacency matrix. Here we find the Laplacian characteristic polynomial of Z n , the group of all integers modulo n. If G is a cyclic group of order n then G ≃ Z n which implies that P s (G) ≃ P s (Z n ). Thus we get Laplacian spectrum of every finite cyclic group. Also we have characterized the same for any finite non-cyclic group. All these details on the Laplacian spectrum are given in Section 2. Moreover we have derived an explicit formula for the permanent of the Laplacian matrix of strong power graphs of any finite group in Section 3. Let A = (a ij ) be a square matrix of order n, then the permanent of A is denoted by per(A) = σ∈Sn a 1σ(1) a 2σ(2) · · · a nσ(n) , where S n is the set of all permutations of 1, 2, · · · , n. The Permanent theory has many role in probability and statistics, and from the theory of permanent of nonnegative matrices several applications of Alexandroff's inequality are illustrated. We refer to [14] and [15] for more on permanents. Also we refer to [11] for group theoretic background.
2 Laplacian spectrum of the strong power graphs of finite groups First we find the characteristic polynomial of the Laplacian matrix associated with the strong power graph P s (Z n ) of the cyclic group Z n .
Proof. Let s i , i = 1, 2, · · · , m = n − φ(n) − 1 be the non generators of Z n . We index the rows and columns of the Laplacian matrix L(P s (Z n )) in order by the non generatorss i (i = 1, 2, · · · m) and the generators of Z n and0 is in last position. Then L(P s (Z n )) = Each row and column sum of the above matrix is zero. Then the characteristic polynomial of Multiply the first row of Θ(L(P s (Z n )), x) by (x − 1) and apply the row operation R Then expanding the resulting determinant through the first row we get, where Θ(Ls 1 ,s 2 ,··· ,s i (P s (Z n )), x) is the determinant obtained form Θ(L(P s (Z n )), x) deleting the rows and columns corresponding to the non generatorss 1 ,s 2 , · · · ,s i . Similarly multiplying the first row and expanding through the first row we get and so Continuing in this way we get with respect to last row we get .
Let G be a cyclic group of order n, then G is isomorphic to Z n ; and the strong power graphs P s (G) and P s (Z n ) of G and Z n respectively are isomorphic. Hence the graphs P s (G) and P s (Z n ) have the same Laplacian spectrum. So by Theorem 2.1 we have: If G is a cyclic group of order n,then the laplacian spectrum of P s (G) is For any non cyclic group G, the strong power graph P s (G) is complete [5]. So their Laplacian spectrum is given by : Let G be a noncyclic group of order n, then the Laplacian spectrum of P s (G) is The algebraic connectivity of a graph Γ, denoted by a(Γ), is the second smallest Laplcian eigen value of Γ [6]. Now the algebraic connectivity has received special attention due to its huge applica- 1 If G is a cyclic group then a(P s (G)) = n − φ(n) − 1.
2 If G is a noncyclic group then a(P s (G)) = n.
Another important application of Laplacian spectrum is on the number of spanning trees of a graph. A spanning tree T of a graph Γ is a subgraph which is a tree having same vertex set is same as Γ. If λ 1 ≥λ 2 ≥λ 3 ≥· · · ≥λ n =0 are the Laplacian eigenvalues of a graph Γ of n-vertices, then the number of spanning trees of Γ is denoted by τ (Γ) is given by 1 If G is a cyclic group then τ (P s (G)) = n n−φ(n)−2 (n − φ(n) − 1)(n − 1) φ(n)−1 .
2 If G is a noncyclic group then τ (P s (G)) = n n−2 .
The graph energy is defined in terms of the spectrum of the adjacency matrix. Depending on the well-developed spectral theory of the Laplacian matrix, recently Gutman et. al [7] have defined the Laplacian energy of a graph Γ with n vertices and m edges as: LE(Γ) = n i=1 |λ i − 2m n |, where λ 1 ≥λ 2 ≥λ 3 ≥· · · ≥λ n =0 are the Laplacian eigen values of the graph Γ. This definition has been adjusted so that the Laplcian energy becomes equal to the energy for any regular graph. For various properties of Laplacian energy we refer [8], [10], [9]. From Proposition 2.2 and Proposition 2.3 we have Corollary 2.6. Let G be a finite group of order n.

Permanent of the Laplacian of strong power graph
Let us recall the definition of permanent of a square matrix. For any square matrix A = (a ij ) of order n, the permanent of A is denoted by per(A) and defined by per(A) = σ∈Sn a 1σ(1) a 2σ(2) · · · a nσ(n) . It is quite difficult to determine the permanent of a square matrix. In this section we have determined the permanent of the Laplacian matrix of strong power graph of any finite group explicitly. Our method is based on the following observation. Let A =(a ij ) be a matrix of order n, then per(A) is equal to the coefficient of x 1 x 2 · · · x n in the expression (a 11 x 1 + a 12 x 2 + · · · a 1n x 1 n)(a 21 x 1 + a 22 x 2 + · · · a 2n x n ) · · · (a n1 x 1 + a n2 x 2 + · · · a nn x n ). Throughout the rest of this section we make the following convention: for any n-functions f 1 and for any n variables x 1 , x 2 , · · · x n , the coefficient of x n 1 x n 2 · · · x n k in a polynomial F (x 1 , x 2 , · · · x n ) will be denoted by C xn 1 xn 2 ···xn k (F (x 1 , x 2 , · · · x n )). Here we first find the permanent of adjacency matrix of any graph with m + n + 1 vertices such that m + n vertices form a clique and the rest vertex is adjacent with n vertices. Let us call such graphs type m + n + 1 graph.

Now We have:
Lemma 3.1. The permanent of adjacency matrix of any graph of type m + n + 1 is Proof. The required permanent is the coefficient of Hence the permanent of the adjacency matrix of the stated graph is Let G be a cyclic group of order n. Then G has m = φ(n) generators none of which is adjacent to the identity element e of G. Thus the set G \ e of all n − 1 non-identity vertices forms a clique and identity e is adjacent to each of the n − φ(n) − 1 non-identity non-generators. Hence from Lemma

it follows immediately that:
Theorem 3.2. Let G be a cyclic group of order n, then the permanent of adjacency matrix of strong power graph of G is Now we compute the permanent of the Laplacian matrix of a graph and hence compute the permanent of Laplacian matrix of strong power graph of any finite group.
Lemma 3.3. The permanent of the Laplcian matrix of any graph Γ of type m + n + 1 is is a product of some (d+1) and (d+2) which is clear from the context. So C x 1 x 2 ···x m+n ((d−m+ i . Now proceeding similarly as in the proof of Hence the permanent of the Laplacian matrix of Γ is The strong power graph of any cyclic group of order n is a graph of type m + n + 1. So we have: For any noncyclic group G of order n the strong power graph P s (G) is complete. Hence we have: Theorem 3.5. The permanent of the Laplacian matrix of the strong power graph P s (G) of a noncyclic group G of order n is (−1) n n!(1 − n 1! + n 2 2! − n 3 3! + · · · + (−1) n n n n! ).