arXiv:1706.02663v2 [math.CO] 13 Jun 2017

arXiv:1706.02663v2 [math.CO] 13 Jun 2017

THE LAPLACIAN SPECTRUM OF POWER GRAPHS OF CYCLIC AND DICYCLIC GROUPS

RAMESH PRASAD PANDA Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, India Abstract. The power graph of a group G is the graph whose vertex set is G and two distinct vertices are adjacent if one is a power of the other. In this paper, the Laplacian spectrum of power graphs of cyclic groups is discussed and certain upper and lower bounds of their algebraic connectivity are given. Then the Laplacian spectrum of power graphs of dicyclic groups is studied and the complete Laplacian spectrum of power graphs of generalized quaternion groups (dicyclic 2-groups) is computed.

1. Introduction All graphs considered in this paper are undirected and simple (i.e., without loops or multiple edges) unless specified otherwise. Kelarev and Quinn [12] introduced the notion of the directed power graph of a Ñ Ý semigroup S as the directed graph G pSq with vertex set S and there is an arc from a vertex u to another vertex v if v “ uα for some α P N. Followed by this, Chakrabarty et al. [4] defined power graph GpSq of a semigroup S as the graph with vertex set S and distinct vertices u and v are adjacent if v “ uα for some α P N or u “ v β for some β P N. Since the introduction, researchers have not only investigated power graphs, but also have shown their usefulness in characterizing finite groups. In [3], Cameron and Ghosh showed that two finite abelian groups with isomorphic power graphs are isomorphic. Cameron [2] proved that if two finite groups have isomorphic power graphs, then their directed power graphs are also isomorphic. Curtin and Pourgholi [7] showed that among all finite groups of a given order, the cyclic group of that order has the maximum number of edges. The proper power graph G ˚ pGq a group G is the graph obtained by removing identity element from power graph of G. In [1, 9], the components of proper power graphs of different groups. For more interesting results on power graphs, the reader is encouraged to read [14, 15, 19]. For a graph Γ with ordered vertex set tv1 , v2 , . . . , vn u, the Laplacian matrix LpΓq of Γ is defined as LpΓq “ DpΓq ´ ApΓq, where DpΓq is the diagonal matrix whose pi, iqth entry is the degree of the vertex vi and ApΓq is the matrix with pi, jqth entry 1 if vi is adjacent to vj and 0 otherwise. ApΓq is called the Adjacency matrix of Γ. E-mail address: [email protected]. 1991 Mathematics Subject Classification. 05C50, 05C25. Key words and phrases. Power graph, Laplacian spectrum, algebraic connectivity, cyclic group, dicyclic group, generalized quaternion group. 1

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Since LpΓq is symmetric, positive semidefinite (cf. [16]), its eigenvalues are real and non-negative. Furthermore, as LpΓq has zero row and colums sums, it is singular and consequently, its smallest eigenvalue is 0. The eigenvalues (spectrum) of LpΓq are (is) called the Laplacian eigenvalues (spectrum) of Γ and are (is) denoted as λ1 pΓq ě λ2 pΓq ě . . . ě λn pΓq “ 0 arranged in non-increasing order and repeated according to their multiplicity. Fiedler [11] proved that λn´1 pΓq ą 0 if and only if Γ is connected and termed λn´1 pΓq as the algebraic connectivity of Γ. More results on Laplacian spectrum of graphs can be found in the text [8]. Chattopadhyay and Panigrahi [5] studied Laplacian spectrum and algebraic connectivity of cyclic and dihedral groups. They showed that for any group G of order n, λ1 pGq “ n. They proved that λn´1 pGpZn qq ě φpnq ` 1 and supplied upper bounds of λn´1 pGpZn qq for all n having two prime factor and product of three primes. Moreover, they presented Laplacian spectrum of GpDn q is terms of that of GpZn q and found that λ2n´1 pGpDn qq “ 1. Mehranian et al. [13] obtained the spectrum of power graphs of cyclic groups, dihedral groups, elementary abelian groups of prime power order and the Mathieu group M11 . In this paper, we study Laplacian spectrum of power graphs of cyclic and dicyclic groups. In Section 3, we find the multiplicity of λn´1 pGpZn qq as an Laplacian eigenvalue of GpZn q. Then we supply upper bound of λn´1 pGpZn qq for all n and lower bound of λn´1 pGpZn qq for certain n. In Section 4, we express characteristic polynomial of LpGpQn qq in terms of LpGpZ2n qq. We obtain certain Laplacian eigenvalues of GpQn q along with their multiplicity and give lower and upper bound of its algebraic connectivity. We find the Laplacian spectrum and hence algebraic connectivity of GpQn q when Qn is generalized quaternion. 2. Preliminaries In a group G, the cyclic subgroup generated by x P G is denoted by xxy. For a prime p, a p-group is a finite group whose order is some power of p. The additive group of integers modulo n is denoted as Zn “ t0, 1, . . . , n ´ 1u. Since any cyclic group of order n is isomorphic to Zn , their corresponding power graphs are also isomorphic. For a positive integer n, the number of positive integers that do not exceed n and are relatively prime to n is denoted by φpnq. The function φ is known as Euler’s phi function. Consider a graph Γ. If vertices u and v are adjacent in Γ, we write u „ v. If Γ is finite, we denote the characteristic polynomial detpxI ´ LpΓqq of LpΓq by ΘpΓ, xq and call it the Laplacian characteristic polynomial of Γ. We now state some existing results which we require subsequently in Section 3 and Section 4. Theorem 2.1 ([4]). Let G be a finite group. (i) The power graph GpGq is always connected. (ii) The power graph GpGq is complete if and only if G is a cyclic group of order 1 or pm , for some prime number p and for some m P N. Theorem 2.2 ([8, Theorem 7.1.2]). For a finite graph Γ, the multiplicity of 0 as an eigenvalue of LpΓq is equal to the number of components of Γ. ` ˘ Theorem 2.3 ([16, Theorem 3.6]). For a graph Γ with n vertices, λn Γ “ 0, and ` ˘ λk Γ “ n ´ λn´k pΓq for 1 ď k ď n ´ 1.

LAPLACIAN SPECTRUM OF POWER GRAPHS OF CYCLIC AND DICYCLIC GROUPS

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Theorem 2.4 ([11]). For a finite graph Γ, λ1 pΓq “ max λ1 pΓi q, where Γ1 , . . . , Γr 1ďiďr

are components of Γ. Theorem 2.5 ([16, Theorem 2.2]). If Γ is a graph with n vertices, then λ1 pΓq ď n. Equality holds if and only if Γ is not connected. 3. Laplacian spectrum of GpZn q In this section, we supply certain upper and lower bounds of algebraic connectivity of GpZn q. It was shown in [5, Theorem 2.12] that λn´1 pGpZn qq ě φpnq ` 1, and equality holds if n is a prime or a product of two distinct primes. We next show that for the equality to hold, this condition is necessary as well. Theorem 3.1. For an integer n ą 1, λn´1 pGpZn qq “ φpnq ` 1 if and only if n is a prime or a product of two distinct primes. Proof. As already mentioned, if n is a prime or a product of two distinct primes, then λn´1 pGpZn qq “ φpnq ` 1. We ´ now¯prove the converse. Let λn´1 pGpZn qq “ φpnq ` 1. Then by Theorem 2.3, λ1 GpZn q “ n ´ φpnq ´ 1. Easily, n ´ φpnq ´ 1 “ 0 if and only if n is a prime.

So let n ´ φpnq ´ 1 ą 0. Then n is not a prime. Moreover, n is not a prime power with power at least two, because this will imply that λn´1 pGpZn qq “ n ‰ φpnq ` 1. Thus n has at least two distinct prime factors. If possible suppose n is not a product of two primes. Then Γ “ GpZn q ´ S is connected, where S is the set consisting of 0 and generators of Zn (cf. [17, Proposition 2.4]). From Theorem 2.4 and the fact that Γ “ GpZn q ´ S is connected (cf. [5, Lemma 2.11]), we have λ1 pGpZn qq “ λ1 pΓq. Since Γ is connected, it follows from Theorem 2.5 that λ1 pΓq ă n´ φpnq´ 1. As a result, λ1 pGpZn qq ă n´ φpnq´ 1, which is a contraction. Hence the proof follows.

From [5, Corollary 2.4] for any integer n ą 1, the multiplicity of n as an eigenvalue of GpZn q is at least φpnq`1. However, we next show that it is exactly φpnq`1 when n is not a prime power. Theorem 3.2. If the integer n ą 1 is not a prime power, then the multiplicity of n as an Laplacian eigenvalue of GpZn q is φpnq ` 1. Proof. Suppose the set S consists of 0 and generators of Zn . Then any vertex in S is adjacent to all other vertices of GpZn q, and |S| “ φpnq ` 1. Moreover, it was shown in [5, Lemma 2.11] that GpZn q ´ S is connected when n is not a prime power. Thus GpZn q has exactly φpnq ` 2 components. Consequently, by Theorem 2.2, the multiplicity of 0 as an eigenvalue of Laplacian eigenvalue of GpZn q is φpnq ` 2. It follows from Theorem 2.3 that the multiplicity of n as an eigenvalue of GpZn q is equal to one less than the multiplicity of 0 as an eigenvalue of GpZn q. Hence the proof follows. αr 1 α2 Theorem 3.3. Let n “ pα 1 p2 . . . pr , where r ě 2, p1 ă p2 ă ¨ ¨ ¨ ă pr are primes and αi P N for 1 ď i ď r, and n is not a product of two distinct primes. If αr “ 1,

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or αr ě 2 and (1)

˙ ˆ 1 1 ě , then 1´ pi 2 i“1

r´1 ś

n r ´1 λn´1 pGpZn qq ď φpnq ` ´ pα φ r pr

ˆ

n r pα r

˙

,

otherwise (2)

λn´1 pGpZn qq ď φpnq `

n `φ r pα r

ˆ

n r pα r

˙

pprαr ´1 ´ 2q.

Proof. Let the right hand side of (1) and (2) be ξ1 pnq and ξ2 pnq, respectively. It was shown in [17] that ξ1 pnq and ξ2 pnq are upper ˆbounds ˙of κpGpZn qq. Furthermore, r´1 ś 1 1 ξ1 pnq ď ξ2 pnq if αr “ 1, or αr ě 2 and 1´ ě , and ξ1 pnq ą ξ2 pnq p 2 i i“1 otherwise. Thus the proof follows from this and the fact that if Γ be a graph on n vertices and is not complete, then λn´1 pΓq ď κpΓq [11]. by Fiedler [11] that if Γ is a graph on n vertices, then λn´1 pΓq ě Ít was shown π¯ 1 2 1 ´ cos κ pΓq. Moreover, if G is a finite group, then κ1 pGpGqq “ δpGpGqq (cf. n [18, Theorem 3.2]). Hence we have the following lemma. ´ π¯ Lemma 3.4. If G a group of order n ě 2, then λn´1 pGpGqq ě 2 1 ´ cos δpGpGqq. n Theorem 3.5. Let p1 ă p2 ă p3 ă p4 be prime numbers and α1 , α2 P N. 1 α2 (i) If n “ pα 1 p2 , then

´ ( π¯ α1 α1 ´1 2 1 λn´1 pGpZn qq ě 2 1 ´ cos qppα ppα 2 ´ 1q ` p1 ´ 1 . 1 ´ p1 n ´ π¯ tφpnq ` p1 p2 ´ 1u. (ii) If n “ p1 p2 p3 , then λn´1 pGpZn qq ě 2 1 ´ cos n 2pp3 ´ 1q (iii) Let n “ p1 p2 p3 p4 . If n is odd or p4 ě p3 ` , then p2 ´ 1 ´ π¯ tφpnq ` p1 p2 ´ 1u, λn´1 pGpZn qq ě 2 1 ´ cos n ´ ¯ π otherwise λn´1 pGpZn qq ě 2 1 ´ cos tpp2 ´ 1qpp3 p4 ` 1q ` 1u. n ´ π¯ Proof. We denote right hand sides of inequalities in (i) and (ii) by 2 1 ´ cos ηk pnq, n k “ 1, 2, respectively. ´ Also, we ¯denote right hand sides of first and second inπ ηk pnq, k “ 3, 4, respectively. It was shown in equalities in (iii) by 2 1 ´ cos n 1 α2 [18, Theorem 4.7] that for n “ pα 1 p2 , δpGpZn qq “ η1 pnq and for n “ p1 p2 p3 , δpGpZn qq “ η2 pnq. For n “ p1 p2 p3 p4 , it was also shown that if n is odd or 2pp3 ´ 1q , then δpGpZn qq “ η3 pnq, otherwise δpGpZn qq “ η4 pnq. Thus the p4 ě p3 ` p2 ´ 1 proof follows from these results and Lemma 3.4.


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4. Laplacin spectrum of GpQn q For an integer n ě 2, the dicyclic group Qn [10] is a finite group of order 4n having presentation @ D (3) Qn “ a, b | a2n “ e, an “ b2 , ab “ ba´1

where e is the identity element of Qn . We first show by induction that pai bq2 “ an for all 0 ď i ď 2n ´ 1. As b2 “ an , it is trivially true for i “ 0. Let it be true for i “ k, where 0 ď k ď 2n ´ 2. Then for i “ k ` 1, pak`1 bq2 “ ak`1 bak`1 b “ ak ba´1 ak`1 b “ pak bq2 “ an , by induction hypothesis. Now for 0 ď i ď n ´ 1, pai bq3 “ an ai b “ anì b and panì bq3 “ an anì b “ ai b. So we get (4)

xai by “ xanì by “ te, ai b, an , anì bu for all 0 ď i ď n ´ 1.

Since b2 “ an , any element of Qn ´ xay can be written as ai b for some 0 ď i ď 2n ´ 1. Thus we conclude from (4) that (5)

Qn “ xay Y

n´1 ď

xai by

i“0

For an integer α ě 2, group having presentation (6)

A E α α´1 Q2α´1 “ a, b | a2 “ e, a2 “ b2 , ab “ ba´1

is called the generalized quaternion group of order 2α`1 [6]. Notice that a generalized quaternion group is in fact a dicyclic group whose order is a power of 2. We fix the following indexing of rows and columns. The first 2n rows and columns of LpGpQn qq are indexed corresponding to the ordered set of vertices pe, an , a, an`1 , . . . , an´1 , a2n´1 q, and last 2n rows and columns are indexed corresponding to the ordered set of vertices pb, an b, ab, an`1 b, . . . , an´1 b, a2n´1 bq. Moreover, the rows and columns of LpGpZ2n qq are indexed corresponding to ordered set p0, n, 1, n ` 1, . . . , n ´ 1, 2n ´ 1q. It was shown in [18, Theorem 5.4] that for n ě 2, δpGpQn qq “ 3. From this and Lemma 3.4, we have the following theorem. Theorem 4.1. For n ě 2, the algebraic connectivity of GpQn q satisfies ´ any integer π¯ λ4n´1 pGpQn qq ě 6 1 ´ cos . 4n Lemma 4.2. For any integer n ě 2, an is adjacent to all other vertices of GpQn q if and only if Qn is generalized quaternion. Proof. Let Qn be generalized quaternion, i.e., n is a power of 2. Then, by Theorem 2.1(ii), xay is a clique, and hence an is adjacent to all other elements of xay in GpQn q. Furthermore, from (4), an is adjacent to ai b for all 0 ď i ď 2n. Thus the proof follows.

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On the other hand, if Qn is not a generalized quaternion group, then there exists 2n a prime factor p ą 2 of n. So a p is an element of order p in xay. Whereas, order 2n of an is 2. Thus an is not adjacent to a p , since this will imply that either 2|p or p|2 neither which is possible. Theorem 4.3. For any integer n ě 2, as an Laplacian eigenvalue of GpQn q, 4n has multiplicity 2 if Qn is generalized quaternion, and 1 otherwise. Proof. As observed earlier, an is adjacent to every element of Qn ´ xay. Moreover, GpQn q ´ te, an u is connected since each element of GpQn q ´ te, an u is adjacent to every element of xay ´ te, an u. So it follows from Lemma 4.2 that the number of components of GpQn q is 3 if n is a power of 2, and 2 otherwise. Accordingly, by Theorem 2.2, the multiplicity of 0 as an Laplacian eigenvalue of GpQn q is 3 if n is a power of 2, and 2 otherwise. By Theorem 2.3, the multiplicity of 4n as an eigenvalue of GpQn q is equal to one less than the multiplicity of 0 as an eigenvalue of GpQn q. Consequently, the result follows. Theorem 4.4. For any integer n ě 2, ΘpGpQn q, xq “ px ´ 2qn px ´ 4qn det pxI2n ´ R2n pxq ´ LpGpZ2n qqq , where R2n pxq is a 2n ˆ 2n matrix given by ¨ ˚ ˚ ˚ R2n pxq “ ˚ ˚ ˝

2n px´2q 2n px´2q

2n `

2n px´2q 2n 2n ` px´2q

0 .. .

0 .. .

0

0

0 ¨¨¨ 0 ¨¨¨ 0 ¨¨¨ .. . . . . 0 ¨¨¨

˛ 0 0‹ ‹ 0‹ ‹. .. ‹ .‚

0

Proof. By adjacency relations of GpZ2n q and GpQn q, the Laplacian matrix of GpQn q is ˙ ˆ LpGpZ2n qq ` M2n N2n , LpGpQn qq “ T N2n P2n where M2n , N2n and P2n are 2n ˆ 2n matrices: (i) The p1, 1q and p2, 2q entries of M2n are both 2n, and all other entries are 0, (ii) N2n has all entries ´1 in first two rows and the rest of the entries are 0, and (iii) P2n is given by ¨ ˛ 3 ´1 0 0 ¨¨¨ 0 ˚´1 3 0 0 ¨¨¨ 0‹ ˚ ‹ ˚ 0 ‹ 0 3 ´1 ¨ ¨ ¨ 0 ˚ ‹ ‹. 0 0 ´1 3 ¨ ¨ ¨ 0 P2n “ ˚ ˚ ‹ ˚¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ‹ ˚ ‹ ˝ 0 ¨¨¨ ¨¨¨ 0 3 ´1‚ 0 ¨¨¨ ¨¨¨ 0 ´1 3 If A, ˆ B, C, D ˙are square matrices of the same order and D is invertible, then A B “ detpDq detpA ´ BD´1 Cq (cf. [20, p. 5]). Therefore, since P2n is det C D invertible, the characteristic polynomial of LpGpQn qq in variable x is given by


7

ˆ

˙ Ń2n xI2n ´ LpGpZ2n qq ´ M2n ΘpGpQn q, xq “ det T Ń2n xI2n ´ P2n ˘ ` T (7) . “ detpxI2n ´ P2n q det xI2n ´ LpGpZ2n qq ´ M2n ´ N2n pxI2n ´ P2n q´1 N2n Observe that

(8)

" ˆ ˙*n x´3 1 “ px ´ 2qn px ´ 4qn . detpxI2n ´ P2n q “ det 1 x´3

Moreover, ¨

pxI2n ´ P2n q´1

x ´ 3 ´1 ˚ ´1 x ´ 3 ˚ ˚ 0 0 ˚ 1 ˚ 0 “ 0 px ´ 3q2 ´ 1 ˚ ˚ ¨¨¨ ¨¨¨ ˚ ˝ 0 ¨¨¨ 0 ¨¨¨

0 0 0 0 x ´ 3 ´1 ´1 x ´ 3 ¨¨¨ ¨¨¨ ¨¨¨ ¨¨¨ ¨¨¨ ¨¨¨

˛ ¨¨¨ 0 ¨¨¨ 0 ‹ ‹ ¨¨¨ 0 ‹ ‹ ¨¨¨ 0 ‹ ‹, ¨¨¨ ¨¨¨ ‹ ‹ x ´ 3 ´1 ‚ ´1 x ´ 3

and hence

T N2n pxI2n ´ P2n q´1 N2n

¨

2npx ´ 4q 2npx ´ 4q 0 ˚2npx ´ 4q 2npx ´ 4q 0 ˚ 1 ˚ 0 0 0 “ px ´ 3q2 ´ 1 ˚ ˝ ¨¨¨ ¨¨¨ ¨¨¨ 0 0 0 ¨

1 1 ˚1 1 2n ˚ ˚0 0 “ ˚ px ´ 2q ˚ .. .. ˝. . 0 0

0 ¨¨¨ 0 ¨¨¨ 0 ¨¨¨ .. . 0 ¨¨¨

¨¨¨ ¨¨¨ ¨¨¨ ¨¨¨ ¨¨¨

˛ 0 0‹ ‹ 0‹ ‹ ¨ ¨ ¨‚ 0

˛ 0 0‹ ‹ 0‹ ‹. .. ‹ .‚ 0

T Thus M2n ` N2n pxI2n ´ P2n q´1 N2n “ R2n pxq. Consequently, from (7) and (8), we have

ΘpGpQn q, xq “ px ´ 2qn px ´ 4qn det pxI2n ´ R2n pxq ´ LpGpZ2n qqq . Corollary 4.5. For any integer n ě 2, 2 and 4 are Laplacian eigenvalues of GpQn q with multiplicities at least n ´ 1 and n, respectively. Proof. Following Theorem 4.4, let T2n pxq be the matrix xI2n ´ R2n pxq ´ LpGpZ2n qq with first row subtracted from second row. Then px´2q detpT2n pxqq is a polynomial and detpT2n pxqq “ detpxI2n ´ Rn pxq ´ LpGpZ2n qqq. Hence the proof follows. Theorem 4.6. For any integer n ě 2, the algebraic connectivity of GpQn q satisfies (9)

1 ă λ4n´1 pGpQn qq ď 2.

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Proof. from ¯ ! Corollary ¯4.5 that λ4n´1 ) pGpQn´qq ď 2. ¯ By Theorem 2.4, ´ It follows ´ ˚ λ1 GpQn q “ max λ1 G pQn q , λ1 pGpteuqq “ λ1 G ˚ pQn q ď 4n ´ 1. Since ¯ ´ G ˚ pQn q is connected, it follows from Theorem 2.5 that λ1 GpQn q ă 4n ´ 1. Consequently, by Theorem 2.3, λ4n´1 pGpQn qq ą 1.

Based on our observation, we state the following. Conjecture 4.7. For any integer n ě 2, 2 and 4 are Laplacian eigenvalues of GpQn q such that (i) 2 has multiplicity n if n is power of 2, and n ´ 1 otherwise. (ii) 4 has multiplicity n for all n ą 2, and its multiplicity 3 for n “ 2. We next obtain the complete Laplacian spectrum of GpQ2α´1 q . In the process, we can observe that Conjecture 4.7 holds true and equality holds in (9) when Qn is generalized quaternion. Theorem 4.8. For an integer α ě 2, the Laplacian eigenvalues of GpQ2α´1 q are 0, 2, 4, 2α and 2α`1 with multiplicities 1, 2α´1 , 2α´1 , 2α ´ 3 and 2, respectively. Proof. From Theorem 4.4, α´1

Θ pGpQ2α´1 q, xq “ px ´ 2q2

α´1

px ´ 4q2

det pxI2α ´ R2α pxq ´ LpGpZ2α qqq .

By Theorem 2.1(ii), GpZ2α q is a complete graph. So the diagonal entries of LpGpZ2α qq are all 2α ´ 1 and the non-diagonal entries are all ´1. Hence we have det pR2α pxq ´ LpGpZ2α qqq x ´ p2α`1 ´ 1q ´ 2α px´2q 2α 1 ´ px´2q 1 “ .. . 1

1´

2α px´2q

x ´ p2α`1 ´ 1q ´ 1 .. . 1

2α px´2q

1 ¨¨¨ 1 ¨¨¨ x ´ p2α ´ 1q ¨ ¨ ¨ .. .. . . 1 ¨¨¨

Multiplying first row by x ´ 1, and then subtracting it from rows, we have xpx ´ 2α`1 q 0 0 α 2α α`1 1´ 2 x ´ p2 ´ 1q ´ 1 px´2q px´2q 1 α 1 1 x ´ p2 ´ 1q “ px ´ 1q . . . .. .. .. 1 1 1 x ´ p2α`1 ´ 1q ´ 1 xpx ´ 2α`1 q “ .. px ´ 1q . 1

2α px´2q

1 ¨¨¨ x ´ p2α ´ 1q ¨ ¨ ¨ .. .. . . 1 ¨¨¨

α x ´ p2 ´ 1q 1 1 1 .. .

sum of rest of the ¨¨¨ ¨¨¨ ¨¨¨ .. . ¨¨¨

0 1 1 .. .

α x ´ p2 ´ 1q

α x ´ p2 ´ 1q 1 1 .. .


9

Multiplying first row by x ´ 2, and then subtracting it from sum of rest of the rows, we have px ´ 1qpx ´ 2α`1 q 0 ¨¨¨ 0 α 1 x ´ p2 ´ 1q ¨ ¨ ¨ 1 xpx ´ 2α`1 q “ .. .. .. .. px ´ 1qpx ´ 2q . . . . α 1 1 ¨ ¨ ¨ x ´ p2 ´ 1q x ´ p2α ´ 1q ¨ ¨ ¨ 1 1 ¨¨¨ 1 α`1 2 q .. .. .. “ xpx´2 . . . px´2q 1 ¨¨¨ 1 α 1 ¨ ¨ ¨ x ´ p2 ´ 1q Multiplying first row by x ´ 3, and then subtracting it from sum of rest of the rows, we have px ´ 2qpx ´ 2α q 0 ¨¨¨ 0 α α`1 2 1 x ´ p2 ´ 1q ¨ ¨ ¨ 1 xpx ´ 2 q “ .. .. . . . . px ´ 2qpx ´ 3q . . . . α 1 1 ¨ ¨ ¨ x ´ p2 ´ 1q x ´ p2α ´ 1q ¨ ¨ ¨ 1 1 ¨ ¨ ¨ 1 . . xpx´2α qpx´2α`1 q2 . . . . “ . . . px´3q 1 ¨¨¨ 1 α 1 ¨ ¨ ¨ x ´ p2 ´ 1q Multiplying first row by x ´ 4, and then subtracting it from sum of rest of the rows, we have px ´ 3qpx ´ 2α q 0 ¨¨¨ α α α`1 2 1 x ´ p2 ´ 1q ¨¨¨ xpx ´ 2 qpx ´ 2 q “ .. .. .. px ´ 3qpx ´ 4q . . . 1 1 ¨¨¨ x ´ p2α ´ 1q ¨ ¨ ¨ 1 1 ¨ ¨ ¨ 1 . . xpx´2α q2 px´2α`1 q2 . . . . “ . . . px´4q 1 ¨¨¨ 1 α 1 ¨ ¨ ¨ x ´ p2 ´ 1q

α x ´ p2 ´ 1q 0 1 .. .

Continuing this process, we get xpx ´ 2α q2 ´4 px ´ 2α`1 q2 px ´ p2α ´ 2qq α

“

α

“ xpx ´ 2α q2

´3

x ´ p2α ´ 1q 1 α 1 x ´ p2 ´ 1q

px ´ 2α`1 q2

Finally, we obtain α´1

Θ pGpQ2α´1 q, xq “ xpx ´ 2q2

α´1

px ´ 4q2

α

px ´ 2α q2

´3

px ´ 2α`1 q2 .

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RAMESH PRASAD PANDA

Therefore, the Laplacian eigenvalues of GpQ2α´1 q are 0, 2, 4, 2α and 2α`1 with multiplicities 1, 2α´1 , 2α´1 , 2α ´ 3 and 2, respectively. This completes the proof of the theorem. Here is an immediate corollary of Theorem 4.8. Corollary 4.9. For an integer α ě 2, the algebraic connectivity of GpQ2α´1 q is 2. 5. Acknowledgment I am grateful to my thesis supervisor Dr. K. V. Krishna for his valuable comments. References [1] D. Bubboloni, M. A. Iranmanesh, and S. M. Shaker. On some graphs associated with the finite alternating groups. Comm. Algebra, 2017. To appear. [2] P. J. Cameron. The power graph of a finite group, II. J. Group Theory, 13(6):779–783, 2010. [3] P. J. Cameron and S. Ghosh. The power graph of a finite group. Discrete Math., 311(13):1220– 1222, 2011. [4] I. Chakrabarty, S. Ghosh, and M. K. Sen. Undirected power graphs of semigroups. Semigroup Forum, 78(3):410–426, 2009. [5] S. Chattopadhyay and P. Panigrahi. On laplacian spectrum of power graphs of finite cyclic and dihedral groups. Linear and Multilinear Algebra, 63(7):1345–1355, 2015. [6] H. S. M. Coxeter and W. O. Moser. Generators and relations for discrete groups, volume 14. Springer Science & Business Media, 2013. [7] B. Curtin and G. Pourgholi. Edge-maximality of power graphs of finite cyclic groups. Journal of Algebraic Combinatorics, 40(2):313–330, 2014. [8] D. M. Cvetkovi´ c, P. Rowlinson, and S. Simi´ c. An introduction to the theory of graph spectra, volume 75. Cambridge University Press Cambridge, 2010. [9] A. Doostabadi and M. Farrokhi D. Ghouchan. On the connectivity of proper power graphs of finite groups. Communications in Algebra, 43(10):4305–4319, 2015. [10] D. S. Dummit and R. M. Foote. Abstract algebra. Wiley India, New Delhi, 2011. [11] M. Fiedler. Algebraic connectivity of graphs. Czechoslovak mathematical journal, 23(2):298– 305, 1973. [12] A. V. Kelarev and S. J. Quinn. A combinatorial property and power graphs of groups. In Contributions to General Algebra 12, Proceedings of the Vienna Conference, pages 229–236, 2000. [13] Z. Mehranian, A. Gholami, and A. Ashrafi. The spectra of power graphs of certain finite groups. Linear and Multilinear Algebra, 2016. [14] M. Mirzargar, A. Ashrafi, and M. Nadjafi-Arani. On the power graph of a finite group. Filomat, 26(6):1201–1208, 2012. [15] A. R. Moghaddamfar, S. Rahbariyan, and W. J. Shi. Certain properties of the power graph associated with a finite group. J. Algebra Appl., 13(7):1450040, 18, 2014. [16] B. Mohar, Y. Alavi, G. Chartrand, and O. Oellermann. The laplacian spectrum of graphs. Graph theory, combinatorics, and applications, 2(871-898):12, 1991. [17] R. P. Panda and K. V. Krishna. On connectedness of power graphs of finite groups. 2017. Submitted. arXiv:1703.08834. [18] R. P. Panda and K. V. Krishna. On minimum degree, edge-connectivity and connectivity of power graphs of finite groups. 2017. Submitted. arXiv:1705.04122. [19] Y. Shitov. Coloring the Power Graph of a Semigroup. Graphs Combin., 33(2):485–487, 2017. [20] F. Zhang. The Schur complement and its applications, volume 4. Springer Science & Business Media, 2006.