Various energies of some super integral groups

VARIOUS ENERGIES OF SOME SUPER INTEGRAL GROUPS

arXiv:1705.00137v1 [math.GR] 29 Apr 2017

PARAMA DUTTA AND RAJAT KANTI NATH∗ Abstract. In this paper, we obtain energy, Laplacian energy and signless Laplacian energy of the commuting graphs of some families of finite non-abelian groups.

1. Introduction Let A(G) and D(G) denote the adjacency matrix and degree matrix of a graph G respectively. Then the Laplacian matrix and signless Laplacian matrix of G are given by L(G) = D(G) − A(G) and Q(G) = D(G) + A(G) respectively. We write Spec(G), L-Spec(G) and Q-Spec(G) to denote the spectrum, Laplacian spectrum bm and Signless Laplacian spectrum of G. Also, Spec(G) = {αa1 1 , αa2 2 , . . . , αal l }, L-Spec(G) = {β1b1 , β2b2 , . . . , βm } c1 c2 cn and Q-Spec(G) = {γ1 , γ2 , . . . , γn } where α1 , α2 , . . . , αn are the eigenvalues of A(G) with multiplicities a1 , a2 , . . . , al ; β1 , β2 , . . . , βm are the eigenvalues of L(G) with multiplicities b1 , b2 , . . . , bm and γ1 , γ2 , . . . , γn are the eigenvalues of Q(G) with multiplicities c1 , c2 , . . . , cn respectively. Harary and Schwenk [17] introduced the concept of integral graphs in 1974. A graph G is called integral or L-integral or Q-integral according as Spec(G) or L-Spec(G) or Q-Spec(G) contains only integers. One may conf. [3, 6, 1, 19, 21, 27] for various results of these graphs. Depending on various spectra of a graph there are various energies called energy, Laplacian energy and signless Laplacian energy denoted by E(G), LE(G) and LE + (G) respectively. These energies are defined as follows: (1.1)

X

E(G) =

λ∈Spec(G)

(1.2)

LE(G) =

X

µ∈L-Spec(G)

and (1.3)

LE + (G) =

X

|λ|,

µ − 2|e(G|) , |v(G)|

ν∈Q-Spec(G)

ν − 2|e(G|) , |v(G)|

where v(G) and e(G) denotes the set of vertices and edges of G respectively. Let G be a finite non-abelian group with center Z(G). The commuting graph of G, denoted by ΓG , is a simple undirected graph whose vertex set is G \ Z(G), and two distinct vertices x and y are adjacent if and only if xy = yx. Various aspects of commuting graphs of different finite groups can be found in [4, 18, 22, 25]. In [14, 12, 15], Dutta and Nath have computed various spectra of ΓG for different families of finite groups. A finite non-abelian group is called super integral if Spec(ΓG ), L-Spec(ΓG ) and Q-Spec(ΓG ) contain only integers. Various examples of super integral groups can be found in [15]. The energy, Laplacian energy and signless Laplacian energy of ΓG are called energy, Laplacian energy and signless Laplacian energy of G respectively. In this paper, we compute various energies of G for some families of super integral groups. It may be mentioned here that the Laplacian energy of non-commuting graphs of various finite non-abelian groups are computed in [13]. 2010 Mathematics Subject Classification. 20D99, 05C50, 15A18, 05C25. Key words and phrases. commuting graph, spectrum, integral graph, finite group. *Corresponding author. 1

2

P. DUTTA AND R. K. NATH

2. Some Computations In this section, we compute various energies of the commuting graphs of some families of finite non-abelian groups. We begin with some families of groups whose central factors are some well-known groups. Theorem 2.1. Let G be a finite group and ha, b : a5 = b4 = 1, b−1 ab = a2 i. Then E(ΓG ) = 38|Z(G)| − 12,

G Z(G)

LE(ΓG ) =

LE + (ΓG ) =

∼ = Sz(2), where Sz(2) is the Suzuki group presented by

( 732|Z(G)|−228

19 120|Z(G)|2 +122|Z(G)|−38 19

( 712|Z(G)|−228

19 120|Z(G)|2 −530|Z(G)|−190 19

if |Z(G)| ≤ 4

if |Z(G)| > 4

and

if |Z(G)| = 1

if |Z(G)| > 1.

Proof. By [14, Theorem 2], we have Spec(ΓG ) = {(−1)19|Z(G)|−6 , (4|Z(G)| − 1)1 , (3|Z(G)| − 1)5 }.

Therefore, by (1.1), we have

E(ΓG ) =19|Z(G)| − 6 + 4|Z(G)| − 1 + 5(3|Z(G)| − 1) = 38|Z(G)| − 12.

Also, v(ΓG ) = 19|Z(G)| and e(ΓG ) = 4|Z(G)|(4|Z(G)|−1)+15|Z(G)|(3|Z(G)|−1) as ΓG = K4|Z(G)| ⊔ 5K3|Z(G)| . 2 Therefore, 2|e(ΓG )| 4|Z(G)|(4|Z(G)| − 1) + 15|Z(G)|(3|Z(G)| − 1) = |v(ΓG )| 19|Z(G)| 16|Z(G)| − 4 + 45|Z(G)| − 15 61|Z(G)| − 19 = = . 19 19 Note that for any two integers r, s, we have (2.1)

r|Z(G)| + s −

By [15, Theorem 2.2], we have

(19r − 61)|Z(G)| + 19(s + 1) 2|e(ΓG )| = . |v(ΓG )| 19

L-Spec(ΓG ) = {06 , (4|Z(G)|)4|Z(G)|−1 , (3|Z(G)|)15|Z(G)|−5 }. 61|Z(G)|−19 15|Z(G)|+19 2|e(ΓG )| Using (2.1), we have 0 − 2|e(G|) = , and − 4|Z(G)| |v(G)| 19 |v(ΓG )| = 19 ( −4|Z(G)|+19 if |Z(G)| ≤ 4 19 3|Z(G)| − 2|e(ΓG )| = −4|Z(G)| + 19 = 4|Z(G)|−19 |v(ΓG )| 19 if |Z(G)| > 4. 19

Therefore, if |Z(G)| ≤ 4, then by (1.2) we have

366|Z(G)| − 114 (4|Z(G)| − 1)(15|Z(G)| + 19) (15|Z(G)| − 5)(−4|Z(G)| + 19) + + 19 19 19 366|Z(G)| − 114 + 60|Z(G)|2 + 61|Z(G)| − 19 − 60|Z(G)|2 + 305|Z(G)| − 95 = 19 732|Z(G)| − 228 . = 19 If |Z(G)| > 4, then by (1.2) we have LE(ΓG ) =

366|Z(G)| − 114 (4|Z(G)| − 1)(15|Z(G)| + 19) (15|Z(G)| − 5)(4|Z(G)| − 19) + + 19 19 19 2 366|Z(G)| − 114 + 60|Z(G)| + 61|Z(G)| − 19 + 60|Z(G)|2 − 305|Z(G)| + 95 = 19 120|Z(G)|2 + 122|Z(G)| − 38 = . 19

LE(ΓG ) =


3

By [15, Theorem 2.2] we also have Q-Spec(ΓG ) = {(8|Z(G)| − 2)1 , (4|Z(G)| − 2)4|Z(G)|−1 , (6|Z(G)| − 2)5 , (3|Z(G)| − 2)15|Z(G)|−5 }.

Now, using (2.1) we have 2|Γ )| 91|Z(G)|−19 , 8|Z(G)| − 2 − |v(ΓGG )| = 19 53|Z(G)|−19 2|e(Γ )| 6|Z(G)| − 2 − |v(ΓGG)| = 19 by (1.3) we have

( −15|Z(G)|+19 if |Z(G)| = 1 2|e(ΓG )| 19 4|Z(G)| − 2 − |v(ΓG )| = 15|Z(G)|−19 if |Z(G)| > 1, 19 4|Z(G)|+19 2|e(ΓG )| and 3|Z(G)| − 2 − |v(ΓG )| = . Hence, if |Z(G)| = 1, then 19

91|Z(G)| − 19 (4|Z(G)| − 1)(−15|Z(G)| + 19) 265|Z(G)| − 95 + + 19 19 19 (15|Z(G)| − 5)(4|Z(G)| + 19) + 19 712|Z(G)| − 228 . = 19 If |Z(G)| > 1, then by (1.3) we have LE + (G) =

91|Z(G)| − 19 (4|Z(G)| − 1)(15|Z(G)| − 19) 5(53|Z(G)| − 19) + + 19 19 19 (15|Z(G)| − 5)(4|Z(G)| + 19) + 19 120|Z(G)|2 − 530|Z(G)| − 190 . = 19

LE + (G) =

Theorem 2.2. Let G be a finite group such that

G Z(G)

∼ = Zp × Zp , where p is a prime integer. Then

E(ΓG ) = LE(ΓG ) = LE + (ΓG ) = 2(p2 − 1)|Z(G)| − 2(p + 1). Proof. By [12, Theorem 2.1] we have 2

Spec(ΓG ) = {(−1)(p

Therefore, by (1.1), we have

−1)|Z(G)|−p−1

, ((p − 1)|Z(G)| − 1)p+1 }.

E(ΓG ) = (p2 − 1)|Z(G)| − p − 1 + (p + 1)((p − 1)|Z(G)| − 1) = 2(p2 − 1)|Z(G)| − 2(p + 1).

We have, |v(ΓG )| = (p2 − 1)|Z(G)| and ΓG = (p + 1)K(p−1)|Z(G)| . Therefore, 2|e(ΓG )| = (p2 − 1)|Z(G)|((p − 1)|Z(G)| − 1) and so 2|e(ΓG )| = (p − 1)|Z(G)| − 1. |v(ΓG )| By [15, Theorem 2.3], we have 2

L-Spec(ΓG ) = {0p+1 , ((p − 1)|Z(G)|)(p −1)|Z(G)|−p−1 }. 2|e(ΓG )| G )| Now, 0 − 2|e(Γ = (p − 1)|Z(G)| − 1 and (p − 1)|Z(G)| − |v(ΓG )| |v(ΓG )| = 1. Hence, by (1.2), we have

LE(ΓG ) = (p + 1)((p − 1)|Z(G)| − 1) + (p2 − 1)|Z(G)| − p − 1 = 2(p2 − 1)|Z(G)| − 2(p + 1).

By [15, Theorem 2.3], we also have

2

Q-Spec(ΓG ) = {(2(p − 1)|Z(G)| − 2)p+1 , ((p − 1)|Z(G)| − 2)(p −1)|Z(G)|−p−1 }. 2|e(ΓG )| G )| = (p − 1)|Z(G)| − 1 and (p − 1)|Z(G)| − 2 − Now, 2(p − 1)|Z(G)| − 2 − 2|e(Γ |v(ΓG )| |v(ΓG )| = 1. Hence, by (1.3), we have LE + (ΓG ) = (p + 1)((p − 1)|Z(G)| − 1) + (p2 − 1)|Z(G)| − p − 1 = 2(p2 − 1)|Z(G)| − 2(p + 1).

4


As a corollary we have the following result. Corollary 2.3. Let G be a non-abelian group of order p3 , for any prime p, then E(ΓG ) = LE(ΓG ) = LE + (ΓG ) = 2p3 − 4p − 2. Proof. Note that |Z(G)| = p and

G Z(G)

∼ = Zp × Zp . Hence the result follows from Theorem 2.2.

Theorem 2.4. Let G be a finite group such that

G Z(G)

∼ = D2m , for m ≥ 2. Then

(1) E(ΓG ) = (4m − 2)|Z(G)| − 2(m + 1). (2) If m = 2; m = 3 and |Z(G)| = 1, 2; m = 4 and |Z(G)| = 1 then LE(ΓG ) =

(2m3 + 2)|Z(G)| − 4m2 − 2m + 2 . 2m − 1

(3) If m ≥ 3 and |Z(G)| ≥ 3; or m = 4 and |Z(G)| ≥ 2; or m ≥ 5 then LE(ΓG ) =

(2m3 − 6m2 + 4m)|Z(G)|2 + (2m2 − 2m + 2)|Z(G)| − 4m + 2 . 2m − 1

(4) If m = 2 then LE + (ΓG ) = 6|Z(G)| − 6. (5) If m = 3 and |Z(G)| = 1 then LE + (ΓG ) =

(6) If m = 3 (7) If m = 4

(8) If m = 4 (9) If m ≥ 5

16 5 . 2 12|Z(G)| +18|Z(G)|−30 + . and |Z(G)| ≥ 2 then LE (ΓG ) = 5 48|Z(G)|2 + . and |Z(G)| ≤ 6 then LE (ΓG ) = 7 2 . and |Z(G)| > 6 then LE + (ΓG ) = 48|Z(G)| +8|Z(G)|−56 7 (2m3 −6m2 +4m)|Z(G)|2 +(m3 −7m2 +4m)|Z(G)|−2m2 +3m−1 + . then LE (ΓG ) = 2m−1

Proof. By [12, Theorem 2.5], we have Spec(ΓG ) = {(−1)(2m−1)|Z(G)|−m−1 , (|Z(G)| − 1)m , ((m − 1)|Z(G)| − 1)1 }. Therefore, by (1.2), we have E(ΓG ) = (2m − 1)|Z(G)| − m − 1 + m(|Z(G)| − 1) + (m − 1)|Z(G)| − 1 = (4m − 2)|Z(G)| − 2(m + 1). Note that |v(ΓG )| = (2m − 1)|Z(G)| and 2|e(ΓG )| = (m − 1)|Z(G)|((m − 1)|Z(G)| − 1) + m|Z(G)|(|Z(G)| −1) since ΓG = K(m−1)|Z(G)| ⊔ mK|Z(G)| . Therefore, 2|e(ΓG )| (m − 1)((m − 1)|Z(G)| − 1) + m(|Z(G)| − 1) (m2 − m + 1)|Z(G)| − 2m + 1 = = . |v(ΓG )| 2m − 1 2m − 1 Note that for any two integers r, s we have (2.2)

r|Z(G)| + s −

((2r + 1)m − m2 − r − 1)|Z(G)| + 2m(s + 1) − s − 1 2|e(ΓG )| = . |v(ΓG )| 2m − 1

By [15, Theorem 2.5], we have L-Spec(ΓG ) = {0m+1 , ((m − 1)|Z(G)|)(m−1)|Z(G)|−1 , (|Z(G)|)m(|Z(G)|−1) } Therefore, using we have (2.2), 2|e(ΓG )| (m2 −m+1)|Z(G)|−2m+1 , (m − 1)|Z(G)| − 0 − |v(ΓG )| = 2m−1

2|e(ΓG )| |v(ΓG )|

=

(m2 −2m)|Z(G)|+2m−1 2m−1

and


|Z(G)| −

2|e(ΓG )| |v(ΓG )| =

 (3m−m2 −2)|Z(G)|+2m−1   2m−1    (−3m+m     

2

+2)|Z(G)|−2m+1 2m−1

if m = 2; or m = 3 and |Z(G)| = 1, 2; or m = 4 and |Z(G)| = 1

if m = 3 and |Z(G)| ≥ 3; or m = 4 and |Z(G)| ≥ 2; or m ≥ 5. Therefore, if m = 2; or m = 3 and |Z(G)| = 1 or 2; or m = 4 and |Z(G)| = 1 then by (1.2), we have LE(ΓG )

(m + 1)((m2 − m + 1)|Z(G)| − 2m + 1) ((m − 1)|Z(G)| − 1)((m2 − 2m)|Z(G)| + 2m − 1) + 2m − 1 2m − 1 (m(|Z(G)| − 1))((3m − m2 − 2)|Z(G)| + 2m − 1) + 2m − 1 (2m3 + 2)|Z(G)| − 4m2 − 2m + 2 . = 2m − 1 =

If m ≥ 3 and |Z(G)| = 3; or m = 4 and |Z(G)| ≥ 2; or m ≥ 5 then by (1.2) we have LE(ΓG )

(m + 1)((m2 − m + 1)|Z(G)| − 2m + 1) ((m − 1)|Z(G)| − 1)((m2 − 2m)|Z(G)| + 2m − 1) + 2m − 1 2m − 1 (m(|Z(G)| − 1))((−3m + m2 + 2)|Z(G)| − 2m + 1) + 2m − 1 (2m3 − 6m2 + 4m)|Z(G)|2 + (2m2 − 2m + 2)|Z(G)| − 4m + 2 = . 2m − 1

=

By [15, Theorem 2.5], we also have

Q-Spec(ΓG ) ={(2(m − 1)|Z(G)| − 2)1 , ((m − 1)|Z(G)| − 2)(m−1)|Z(G)|−1 ,

(2|Z(G)| − 2)m , (|Z(G)| − 2)m(|Z(G)|−1) }.

Now, using (2.2), we have (3m2 −5m+1)|Z(G)|−2m+1 2|e(Γ )| , 2(m − 1)|Z(G)| − 2 − |v(ΓGG)| = 2m−1 ( 2 (m −2m)|Z(G)|−2m+1 if m = 3 and |Z(G)| ≥ 2; or m ≥ 4 2|e(Γ )| 2m−1 (m − 1)|Z(G)| − 2 − |v(ΓGG)| = (−m2 +2m)|Z(G)|+2m−1 if m = 2; m = 3 and |Z(G)| = 1, 2m−1  (5m−m2 −3)|Z(G)|−2m+1  if m = 2; m = 3 and |Z(G)| ≥ 2;  2m−1    or m = 4 and |Z(G)| > 6 2|e(Γ )| and 2|Z(G)| − 2 − |v(ΓGG)| = (−5m+m2 +3)|Z(G)|+2m−1  if m = 3 and |Z(G)| = 1;  2m−1    m = 4 and |Z(G)| ≤ 6; or m ≥ 5 2|e(ΓG )| (−3m+m2 +2)|Z(G)|+2m−1 . |Z(G)| − 2 − |v(ΓG )| = 2m−1 If m = 2, then by (1.3), we have LE + (ΓG )

(3m2 − 5m + 1)|Z(G)| − 2m + 1 ((m − 1)|Z(G)| − 1)((−m2 + 2m)|Z(G)| + 2m − 1) + 2m − 1 2m − 1 m((5m − m2 − 3)|Z(G)| − 2m + 1) m(|Z(G)| − 1)((−3m + m2 + 2)|Z(G)| + 2m − 1) + + 2m − 1 2m − 1 3|Z(G)| − 3 3(|Z(G)| − 1) 6|Z(G)| − 6 6(|Z(G)| − 1) + + + = 3 3 3 3 =6|Z(G)| − 6. =

5

6


If m = 3 and |Z(G)| = 1, then by (1.3), we have LE + (ΓG ) (3m2 − 5m + 1)|Z(G)| − 2m + 1 ((m − 1)|Z(G)| − 1)((−m2 + 2m)|Z(G)| + 2m − 1) + 2m − 1 2m − 1 m((−5m + m2 + 3)|Z(G)| + 2m − 1) m(|Z(G)| − 1)((−3m + m2 + 2)|Z(G)| + 2m − 1) + + 2m − 1 2m − 1 8 2 6 16 = + + = . 5 5 5 5 =

If m = 3 and |Z(G)| ≥ 2 then, by (1.3), we have LE + (ΓG ) (3m2 − 5m + 1)|Z(G)| − 2m + 1 ((m − 1)|Z(G)| − 1)((m2 − 2m)|Z(G)| − 2m + 1) + 2m − 1 2m − 1 m((5m − m2 − 3)|Z(G)| − 2m + 1) m(|Z(G)| − 1)((−3m + m2 + 2)|Z(G)| + 2m − 1) + + 2m − 1 2m − 1 13|Z(G)| − 5 (2|Z(G)| − 1)(3|Z(G)| − 5) 9|Z(G)| − 15 (3|Z(G)| − 3)(2|Z(G)| + 5) + + + = 5 5 5 5 13|Z(G)| − 5 6|Z(G)|2 − 13|Z(G)| + 5 9|Z(G)| − 15 6|Z(G)|2 + 9|Z(G)| − 15 + + + = 5 5 5 5 12|Z(G)|2 + 18|Z(G)| − 30 . = 5 =

If m = 4 and |Z(G)| ≤ 6 then, by (1.3), we have LE + (ΓG ) (3m2 − 5m + 1)|Z(G)| − 2m + 1 ((m − 1)|Z(G)| − 1)((m2 − 2m)|Z(G)| − 2m + 1) + 2m − 1 2m − 1 m((−5m + m2 + 3)|Z(G)| + 2m − 1) m(|Z(G)| − 1)((−3m + m2 + 2)|Z(G)| + 2m − 1) + + 2m − 1 2m − 1 29|Z(G)| − 7 (3|Z(G)| − 1)(8|Z(G)| − 7) 4(−|Z(G)| + 7) 4(|Z(G)| − 1)(6|Z(G)| + 7) + + + = 7 7 7 7 48|Z(G)|2 = . 7 =

If m = 4 and |Z(G)| > 6 then, by (1.3), we have LE + (ΓG ) (3m2 − 5m + 1)|Z(G)| − 2m + 1 ((m − 1)|Z(G)| − 1)((m2 − 2m)|Z(G)| − 2m + 1) + 2m − 1 2m − 1 m((5m − m2 − 3)|Z(G)| − 2m + 1) m(|Z(G)| − 1)((−3m + m2 + 2)|Z(G)| + 2m − 1) + + 2m − 1 2m − 1 29|Z(G)| − 7 24|Z(G)|2 − 29|Z(G)| + 7 4|Z(G)| − 28 24|Z(G)|2 + 4|Z(G)| − 28 + + + = 7 7 7 7 48|Z(G)|2 + 8|Z(G)| − 56 = . 7 =


7

If m ≥ 5 then, by (1.3), we have LE + (ΓG )

(3m2 − 5m + 1)|Z(G)| − 2m + 1 ((m − 1)|Z(G)| − 1)((m2 − 2m)|Z(G)| − 2m + 1) + 2m − 1 2m − 1 m((−5m + m2 + 3)|Z(G)| + 2m − 1) m(|Z(G)| − 1)((−3m + m2 + 2)|Z(G)| + 2m − 1) + + 2m − 1 2m − 1 3 2 2 3 2 2 (2m − 6m + 4m)|Z(G)| + (m − 7m + 4m)|Z(G)| − 2m + 3m − 1 = . 2m − 1 =

Using Theorem 2.4, we now compute the energy, Laplacian energy and signless Laplacian energy of the commuting graphs of the groups M2mn , D2m and Q4n respectively. Corollary 2.5. Let M2mn = ha, b : am = b2n = 1, bab−1 = a−1 i be a metacyclic group, where m > 2. If m is odd then, E(ΓM2mn ) = (4m − 2)n − 2(m + 1),

LE(ΓM2mn ) = and LE + (ΓM2mn ) = If m is even then,

 56n−40   52 ,

12n +14n−10 , 5   (2m3 −6m2 +4m)n2 +(2m2 −2m+2)n−4m+2 2m−1

if m = 3 and n = 1, 2 ; if m = 3 and n ≥ 3; otherwise,

 16   5 ,2

12n +18n−30 , 5   (2m3 −6m2 +4m)n2 +(m3 −7m2 +4m)n−2m2 +3m−1 2m−1

if m = 3 and n = 1 ; if m = 3 and n ≥ 2; otherwise.

E(ΓM2mn ) = (4m − 4)n − (m + 2),

LE(ΓM2mn ) =

 16n−9   3 ,   72   5, 2

48n +28n−10 , 5   192n2 +52n−14  ,  7   (m3 −6m 2  +8m)n2 +(m2 −2m+4)n−2m+2

if if if if

m = 4; m = 6 and n = 1; m = 6 and n ≥ 2; m = 8 and n ≥ 1 ;

, otherwise ; m−1   if m = 4; 12n2 − 6,   48n +36n−30  , if m = 6 and n ≥ 1;  5  2 , if m = 8 and n ≤ 3; LE + (ΓM2mn ) = 192n 7   192n2 +16n−56  , if m = 8 and n > 3 ;  7    (4m3 −24m2 +32m)n2 +(m3 −14m2 +16m)n−2m2 +6m−4 , otherwise . 4(m−1) m

Proof. Observe that Z(M2mn ) = hb2 i or hb2 i ∪ a 2 hb2 i according as m is odd or even. Also, it is easy to see M2mn ∼ that Z(M = D2m or Dm according as m is odd or even. Hence, the result follows from Theorem 2.4. 2mn ) As a corollary to the above result we have the following results. Corollary 2.6. Let D2m = ha, b : am = b2 = 1, bab−1 = a−1 i be the dihedral group of order 2m, where m > 2. Then If m is odd, then E(ΓD2m ) = 2m − 3,

8


LE(ΓD2m ) =

(

16 5 , 2(m+1)(m−1)(m−2) 2m−1

if m = 3 ; otherwise,

+

LE (ΓD2m ) =

and

(

16 5 , 3m3 −15m2 +11m−1 2m−1

if m = 3 ; otherwise.

If m is even, then E(ΓD2m ) = 3m − 6,

LE(ΓD2m ) =

7  3,    72 , 5

230  ,    m73 −5m2 +4m+6 m−1

if m = 4 ; if m = 6 ; if m = 8 ; otherwise,

and

LE + (ΓD2m ) =

 6,     54 , 5

192   7 ,  3  5m −40m2 +42m−4 4(m−1)

if m = 4 ; if m = 6 ; if m = 8 ; otherwise.

Corollary 2.7. Let Q4m = hx, y : y 2m = 1, x2 = y m , yxy −1 = y −1 i, where m ≥ 2, be the generalized quaternion group of order 4m. Then E(ΓQ4m ) = 6m − 6,

LE(ΓQ4m ) =

 6,     72 , 5

230   7 ,  3  8m −20m2 +8m+6 2m−1

if m = 2 ; if m = 3 ; if m = 4 ; otherwise,

and

LE + (ΓQ4m ) =

 6,     54 , 5

192   7 ,  3  10m −40m2 +27m−1 2m−1

Proof. The result follows from Theorem 2.4 noting that Z(Q4m ) = {1, am } and

Q4m Z(Q4m )

if m = 2 ; if m = 3 ; if m = 4 ; otherwise.

∼ = D2m .

In this section, Now we compute various energies of the commuting graphs of some well-known families of finite non-abelian groups. Proposition 2.8. Let G be a non-abelian group of order pq, where p and q are primes with p | (q − 1). Then  2 q(q −3q−3pq2 +1)  , if p = 2 and q 6= 3 ,  pq−1  2pq(2pq−p−q2 −3q+1)+q(5q2 −6q+4) E(ΓG ) = 2q(p − 1) − 3, LE(ΓG ) = , if p = 2 and q = 3 , pq−1    −2pq(pq−2p−q2 +4)−q(3q2 −6q+2)+4 , otherwise; pq−1 and

+

LE (ΓG ) =

( 2pq(2q−p−1)−(2q2 +3q−6)

, pq−1 2p2 q(1−q)+2q3 (p−1)+q(2q−2p+1)−2 , pq−1

if p = 2 and q = 3 , otherwise.

Proof. By [14, Lemma 3], we have Spec(ΓG ) = {(−1)pq−q−1 , (p − 2)q , (q − 2)1 }. Therefore, by (1.1) we have E(ΓG ) = 2q(p − 1) − 3. Note that |v(ΓG )| = pq − 1 and |e(ΓG )| =

p2 q−3pq+q2 −q+2 2

since ΓG = qKp−1 ⊔ Kq−1 . Therefore,

2|e(ΓG )| p2 q − 3pq + q 2 − q + 2 = . |v(ΓG )| pq − 1 By [15, Proposition 2.9], we have L-Spec(ΓG ) = {0q+1 , (q − 1)q−2 , (p − 1)pq−2q }.


Therefore, 2 2 0 − 2|e(ΓG )| = p q − 3pq + q − q + 2 , |v(ΓG )| pq − 1

and

9

( −pq(q−p)+2q(q−p)+1 , if p = 2 ; 2|e(Γ )| G pq−1 q − 1 − = pq(q−p)−2q(q−p)−1 |v(ΓG )| , otherwise pq−1

( q(2p−q)+(q−p)−1 , if p = 2 and q = 2 ; 2|e(Γ )| G pq−1 = p − 1 − −q(2p−q)−(q−p)+1 |v(ΓG )| , otherwise. pq−1

Hence, by (1.2) we have, if p = 2 and q 6= 3, then

(q + 1)(p2 q − 3pq + q 2 − q + 2) (q − 2)(−pq(q − p) + 2q(q − p) + 1) + pq − 1 pq − 1 (pq − 2q)(−q(2p − q) − (q − p) + 1) + pq − 1 2 2 q(q − 3q − 3pq + 1) = . pq − 1 If p = 2 and q = 3, then LE(ΓG ) =

(q + 1)(p2 q − 3pq + q 2 − q + 2) (q − 2)(−pq(q − p) + 2q(q − p) + 1) + pq − 1 pq − 1 (pq − 2q)(q(2p − q) + (q − p) − 1) + pq − 1 2pq(2pq − p − q 2 − 3q + 1) + q(5q 2 − 6q + 4) . = pq − 1

LE(ΓG ) =

Otherwise,

(q + 1)(p2 q − 3pq + q 2 − q + 2) (q − 2)(pq(q − p) − 2q(q − p) − 1) + pq − 1 pq − 1 (pq − 2q)(−q(2p − q) − (q − p) + 1) + pq − 1 −2pq(pq − 2p − q 2 + 4) − q(3q 2 − 6q + 2) + 4 . = pq − 1

LE(ΓG ) =

By [15, Proposition 2.9], we also have

Q-Spec(ΓG ) = {(2q − 4)1 , (q − 3)q−2 , (2p − 4)q , (p − 3)pq−2q }. 2|e(ΓG )| pq(2q−p)−q(p+q+1)+2 , |v(ΓG )| = pq−1 ( −pq(q−p)+q2 −2 , if p = 2 and q = 3 ; 2|e(Γ )| G pq−1 q − 3 − = pq(q−p)−q2 +2 |v(ΓG )| , otherwise, pq−1 2|e(ΓG )| −pq(p−1)+q(q−1)+2p−2 2|e(ΓG )| and p − 3 − |v(ΓG )| = p+q(q−1)−1 . Therefore, by (1.3) we 2p − 4 − |v(ΓG )| = pq−1 pq−1 have, if p = 2 and q = 3,

Therefore, 2q − 4 −

pq(2q − p) − q(p + q + 1) + 2 (q − 2)(−pq(q − p) + q 2 − 2) + pq − 1 pq − 1 q(−pq(p − 1) + q(q − 1) + 2p − 2) (pq − 2q)(p + q(q − 1) − 1) + + pq − 1 pq − 1 2pq(2q − p − 1) − (2q 2 + 3q − 6) . = pq − 1

LE + (ΓG ) =

10


Otherwise, pq(2q − p) − q(p + q + 1) + 2 (q − 2)(pq(q − p) − q 2 + 2) + pq − 1 pq − 1 q(−pq(p − 1) + q(q − 1) + 2p − 2) (pq − 2q)(p + q(q − 1) − 1) + + pq − 1 pq − 1 2 3 2p q(1 − q) + 2q (p − 1) + q(2q − 2p + 1) − 2 . = pq − 1

LE + (ΓG ) =

This completes the proof.

n−1

Proposition 2.9. Let QD2n denote the quasidihedral group ha, b : a2 n ≥ 4. Then E(ΓQ D2n ) = 3(2n−1 − 2), LE(ΓQ D2n ) = and

= b2 = 1, bab−1 = a2

−1

i, where

23n−3 − 5.22n−2 + 4.2n−1 + 12 2n−1 − 1

LE + (ΓQ D2n ) = Proof. By [14, Proposition 1], we have

n−2

5.23n−4 − 30.22n−3 + 40.2n−2 . 2n−1 − 1 n

Spec(ΓQD2n ) = {(−1)2

−2n−2 −3

n−2

, 12

, (2n−1 − 3)1 }.

Therefore, by (1.1) we have E(ΓQD2n ) = 3(2n−1 − 2).

Note that |v(ΓQD2n )| = 2n−1 − 1 and |e(ΓQD2n )| =

22n−2 −4.2n−1 +6 2

since ΓG = 2n−2 K2 ⊔ K2n−1 −2 . Therefore,

2|e(ΓQD2n )| 22n−2 − 4.2n−1 + 6 = . |v(ΓQD2n )| 2(2n−1 − 1)

By [15, Proposition 2.10], we have

n−2

n−1

n−2

L-Spec(ΓQD2n ) = {02 +1 , (2n−1 − 2)2 −3 , 22 }. 2n−2 n−1 2|e(ΓQD2n )| 2|e(ΓQD2n )| −2 22n−2 −4.2n−1 +6 n−1 = , 2 − 2 − and Therefore, 0 − |v(ΓQD = 2 2(2−2.2 n−1 −1) n−1 −1) 2(2 |v(Γ )| )| G 2n 2|e(ΓQD2n )| 22n−2 −8.2n−1 +10 . Hence, by (1.2) we have 2 − |v(ΓG )| = 2(2n−1 −1) LE(ΓQD2n ) =

By [15, Proposition 2.10], we also have

23n−3 − 5.22n−2 + 4.2n−1 + 12 . 2n−1 − 1 n−1

n−2

n−2

Q-Spec(ΓQD2n ) = {(2n − 6)1 , (2n−1 − 4)2 −3 , 22 , 02 }. 2n−2 n−1 2|e(ΓQD2n )| 2|e(ΓQD )| +6 n−1 22n−2 −6.2n−1 +2 , , − 4 − Therefore, 2n − 6 − |v(ΓQD 2nn)| = 3.2 2(2−12.2 2 n−1 −1) |v(ΓQD2n )| = 2(2n−1 −1) 2 2n−2 n−1 2n−2 n−1 2|e(ΓQD n )| 2|e(ΓQD2n )| +10 +6 and 0 − |v(ΓQD . Therefore, by (1.3) we have 2 − |v(ΓQD 2n )| = 2 2(2−8.2 = 2 2(2−4.2 n−1 −1) n−1 −1) n )| 2

2

3.22n−2 − 12.2n−1 + 6 (2n−1 − 3)(22n−2 − 6.2n−1 + 2 + LE + (ΓQD2n ) = 2(2n−1 − 1) 2(2n−1 − 1) n−2 2n−2 n−1 2n−2 (22n−2 − 4.2n−1 + 6 2 (2 − 8.2 + 10 + + ++ 2(2n−1 − 1) 2(2n−1 − 1) 3n−4 2n−3 n−2 5.2 − 30.2 + 40.2 . = n−1 2 −1


11

Proposition 2.10. Let G denote the projective special linear group P SL(2, 2k ), where k ≥ 2. Then E(ΓG ) = 23k+1 − 22k+1 − 2k+2 − 4, LE(ΓG ) = and +

LE (ΓG ) =

2.26k − 2.25k − 8.24k − 6.23k + 6.22k + 8.2k + 4 , 23k − 2k − 1 (

26k +25k −3.24k −7.23k +4.2k +4 , 23k −2k −1 2.26k −2.25k −8.24k −6.23k +6.22k +8.2k +4 , 23k −2k −1

if k=2; otherwise.

Proof. By [14, Proposition 2], we have 3k

Spec(ΓG ) = {(−1)2

−22k −2k+1 −2

k−1

, (2k − 1)2

(2k −1)

k

, (2k − 2)2

+1

k−1

, (2k − 3)2

(2k +1)

}.

Therefore, by (1.1) we have E(ΓG ) = 23k+1 − 22k+1 − 2k+2 − 4. 24k −2.23k −22k +2.2k +2 2

Note that |v(ΓG )| = 23k − 2k − 1 and |e(ΓG )| = 1)K2k −2 ⊔ 2k−1 (2k − 1)K2k . Therefore,

since ΓG = (2k + 1)K2k −1 ⊔2k−1 (2k +

24k − 2.23k − 22k + 2.2k + 2 2|e(ΓG )| = . |v(ΓG )| 23k − 2k − 1 By [15, Proposition 2.11], we have 2k

2k

k

k

k−1

2k

k+1

k−1

2k

k+1

L-Spec(ΓG ) = {02 +2 +1 , (2k − 1)2 −2 −2 , (2k − 2)2 (2 −2 −3) , (2k )2 (2 −2 +1) }. 2|e(ΓG )| 2|e(ΓG )| 24k −2.23k −22k +2.2k +2 k 23k −2.2k −1 k G )| = = Therefore, 0 − 2|e(Γ , , − 1 − − 2 − 2 2 3k k 3k k |v(ΓG )| |v(ΓG )| = 2 −2 −1 2 −2 −1 |v(ΓG )| k 2|e(ΓG )| 2.23k −3.2k −2 2k and 2 − |v(ΓG )| = 23k −2k −1 . Hence, by (1.2) we have 23k −2k −1 (22k + 2k + 1)(24k − 2.23k − 22k + 2.2k + 2) (22k − 2k − 2)(23k − 2.2k − 1) + 23k − 2k − 1 23k − 2k − 1 k−1 2k k+1 3k k−1 2k k+1 k 2 (2 − 2 + 1)(2.2 − 3.2k − 2) 2 (2 − 2 − 3)2 + + 23k − 2k − 1 23k − 2k − 1 6k 5k 4k 3k 2k k 2.2 − 2.2 − 3.2 − 4.2 + 3.2 + 8.2 + 4 = . 23k − 2k − 1

LE(ΓG ) =

By [15, Proposition 2.11], we also have k

Q-Spec(ΓG ) = {(2k+1 − 4)2 Therefore, 2k+1 − 4 −

2|e(ΓG )| |v(ΓG )| =

+1

2k

, (2k − 3)2

−2k −2

k−1

,(2k+1 − 6)2

k−1

(2k+1 − 2)2

24k −2.23k −22k +2 k , 2 − 3 − 3k k 2 −2 −1

(2k +1) k

(2 −1)

2|e(ΓG )| |v(ΓG )| =

k−1

, (2k − 4)2

k−1

, (2k − 2)2

(22k −2k+1 −3) 2k

(2

k+1

−2

+1)

, }.

23k −1 , 23k −2k −1

( −24k +4.23k +22k −2.2k −4 k+1 , if k=2 ; 2|e(Γ )| G 23k −2k −1 = 2 −6− 4k 3k 2k k 2 −4.2 −2 +2.2 +4 |v(ΓG )| , otherwise, 23k −2k −1 3k k 2|e(Γ )| −2k −2 k+1 24k −22k −2.2k G )| , 2 and 2k − 2 − − 2 − 2|e(Γ 2 − 4 − |v(ΓGG)| = 2.2 |v(ΓG )| = 23k −2k −1 23k −2k −1 Therefore, by (1.3) we have, if k = 2 then

2|e(ΓG )| |v(ΓG )|

=

2k . 23k −2k −1

12


(2k + 1)(24k − 2.23k − 22k + 2) (22k − 2k − 2)(23k − 1) + 23k − 2k − 1 23k − 2k − 1 k−1 k 4k 3k 2k 2 (2 + 1)(−2 + 4.2 + 2 − 2.2k − 4) 2k−1 (22k − 2k+1 − 3)(2.23k − 2k − 2) + + 23k − 2k − 1 23k − 2k − 1 2k−1 (2k − 1)(24k − 22k − 2.2k ) 2k−1 (22k − 2k+1 + 1)2k + + 23k − 2k − 1 23k − 2k − 1 6k 5k 4k 3k k 2 + 2 − 3.2 − 7.2 + 4.2 + 4 . = 23k − 2k − 1

LE + (ΓG ) =

Otherwise,

(2k + 1)(24k − 2.23k − 22k + 2) (22k − 2k − 2)(23k − 1) + 23k − 2k − 1 23k − 2k − 1 k−1 k 4k 3k 2k k 2 (2 + 1)(2 − 4.2 − 2 + 2.2 + 4) 2k−1 (22k − 2k+1 − 3)(2.23k − 2k − 2) + + 23k − 2k − 1 23k − 2k − 1 2k−1 (2k − 1)(24k − 22k − 2.2k ) 2k−1 (22k − 2k+1 + 1)2k + + 23k − 2k − 1 23k − 2k − 1 6k 5k 4k 3k 2k k 2.2 − 2.2 − 8.2 − 6.2 + 6.2 + 8.2 + 4 = . 23k − 2k − 1

LE + (ΓG ) =


Proposition 2.11. Let G denote the general linear group GL(2, q), where q = pn > 2 and p is a prime. Then 2q 4 − 2q 3 − 8q 2 − 5q , 2 2q 9 − 6q 8 + 4q 7 + 8q 6 − 10q 5 + 4q 3 + 4q 2 − 8q and LE(ΓG ) = 2(q − 1)(q 3 − q − 1) q 10 − 4q 9 + 10q 8 + 3q 7 − 23q 6 − 9q 5 + 22q 4 + 10q 3 − 9q 2 − 4q LE + (ΓG ) = . 2(q − 1)(q 3 − q − 1) E(ΓG ) =

Proof. By [14, Proposition 3], we have Spec(ΓG ) = {(−1)q

4

−q3 −2q2 −q

, (q 2 − 3q + 1)

q(q+1) 2

, (q 2 − q − 1)

q(q−1) 2

, (q 2 − 2q)q+1 }.

Therefore, by (1.1) we have E(ΓG ) =

2q 4 − 2q 3 − 8q 2 − 5q . 2

Note that |v(ΓG )| = (q − 1)(q 3 − q − 1) and |e(ΓG )| = ⊔ q(q−1) Kq2 −q ⊔ (q + 1)Kq2 −2q+1 . Therefore, 2

q6 −3q5 +q4 +3q3 −q2 −q 2

as ΓG =

q(q+1) Kq2 −3q+2 2

q 6 − 3q 5 + q 4 + 3q 3 − q 2 − q 2|e(ΓG )| = . |v(ΓG )| (q − 1)(q 3 − q − 1) By [15, Proposition 2.12], we have 2

q(q+1)(q2 −3q+1)

q(q−1)(q2 −q−1)

2 2 L-Spec(ΓG ) = {0q +q+1 , (q 2 − 3q + 2) , (q 2 − q) , (q 2 − 2q + 1)q(q+1)(q−2) }. 2|e(ΓG )| q6 −3q5 +q4 +3q3 −q2 −q 2 q5 −3q4 +2q3 +2q−2 G )| Therefore, 0 − 2|e(Γ = , − 3q + 2 − q 3 |v(ΓG )| (q−1)(q −q−1) |v(ΓG )| = (q−1)(q3 −q−1) , 2 2 2|e(ΓG )| 2|e(ΓG )| q5 −q4 −2q3 +2q2 q4 −2q3 +q2 −q+1 q − q − |v(ΓG )| = (q−1)(q3 −q−1) and q − 2q + 1 − |v(ΓG )| = (q−1)(q3 −q−1) .


13

Hence, by (1.2) we have q(q + 1)(q 2 − 3q + 1) q 5 − 3q 4 + 2q 3 + 2q − 2 (q 2 + q + 1)(q 6 − 3q 5 + q 4 + 3q 3 − q 2 − q) + LE(ΓG ) = (q − 1)(q 3 − q − 1) 2 (q − 1)(q 3 − q − 1) 2 5 4 3 2 4 3 q(q − 1)(q − q − 1) q − q − 2q + 2q q(q + 1)(q − 2)(q − 2q + q 2 − q + 1) + + 2 (q − 1)(q 3 − q − 1) (q − 1)(q 3 − q − 1) 9 8 7 6 5 3 2 2q − 6q + 4q + 8q − 10q + 4q + 4q − 8q . = 2(q − 1)(q 3 − q − 1) By [15, Proposition 2.12], we also have

Q-Spec(ΓG ) ={(2q 2 − 6q − 2)

q(q+1) 2

, (q 2 − 3q)

q(q+1)(q2 −3q+1) 2

q(q−1)(q2 −q−1) 2

, (2q 2 − 2q − 2)

q(q−1) 2

,

, (2q 2 − 4q)q+1 , (q 2 + 2q − 1)q(q+1)(q−2) }. (q 2 − q − 2) 2|e(ΓG )| q6 +7q5 −11q4 −7q3 +5q2 +7q−2 2 q5 −q4 +q2 +2q G )| Therefore, 2q 2 − 6q − 2 − 2|e(Γ = , − 3q − q 3 |v(ΓG )| (q−1)(q −q−1) |v(ΓG )| = (q−1)(q3 −q−1) , 6 5 2 2|e(Γ )| −3q4 +q3 +6q2 −q−2 2 q5 −3q4 +4q2 −2 G )| , q − q − 2 − 2|e(Γ 2q − 2q − 2 − |v(ΓGG)| = q +3q(q−1)(q 3 −q−1) |v(ΓG )| = (q−1)(q3 −q−1) , 6 5 2 2|e(Γ )| +q4 +q3 +3q2 −3q 4q5 −5q4 −4q3 +3q2 +3q−1 G )| and q 2 + 2q − 1 − 2|e(Γ . Therefore, 2q − 4q − |v(ΓGG)| = q −3q (q−1)(q3 −q−1) |v(ΓG )| = (q−1)(q3 −q−1)

by (1.3) we have q(q + 1) q 6 + 7q 5 − 11q 4 − 7q 3 + 5q 2 + 7q − 2 q(q + 1)(q 2 − 3q + 1) q 5 − q 4 + q 2 + 2q + LE (ΓG ) = + 2 (q − 1)(q 3 − q − 1) 2 (q − 1)(q 3 − q − 1) 6 5 5 4 3 2 2 q(q − 1)(q − q − 1) q − 3q 4 + 4q 2 − 2 q(q − 1) q + 3q − 3q + q + 6q − q − 2 + + 2 (q − 1)(q 3 − q − 1) 2 (q − 1)(q 3 − q − 1) 5 4 3 6 5 4 3 2 (q + 1)(q − 3q + q + q + 3q − 3q) q(q + 1)(q − 2)(4q − 5q − 4q + 3q 2 + 3q − 1) + + (q − 1)(q 3 − q − 1) (q − 1)(q 3 − q − 1) 10 9 8 7 6 5 4 3 2 q − 4q + 10q + 3q − 23q − 9q + 22q + 10q − 9q − 4q . = 2(q − 1)(q 3 − q − 1)

Proposition 2.12. Let F = GF (2 ), n ≥ 2 and ϑ be the Frobenius automorphism of F , i. e., ϑ(x) = x2 for all x ∈ F . If G denotes the group     1 0 0   1 0 : a, b ∈ F U (a, b) = a   b ϑ(a) 1 n

under matrix multiplication given by U (a, b)U (a′ , b′ ) = U (a + a′ , b + b′ + a′ ϑ(a)), then 2

E(ΓG ) = LE(ΓG ) = LE + (ΓG ) = 2(2n − 1) . Proof. By [14, Proposition 4], we have n

Spec(ΓG ) = {(−1)(2

−1)2

n

, (2n − 1)2

−1

}.

Therefore, by (1.1) we have 2

E(ΓG ) = 2(2n − 1) .

Note that |v(ΓG )| = 2n (2n − 1) and |e(ΓG )| =


23n −22n+1 +2n 2

since ΓG = (2n − 1)K2n . Therefore,

2|e(ΓG )| = 2n − 1. |v(ΓG )| n

L-Spec(ΓG ) = {02

−1

2n

, (2n )2

−2n+1 +1

}.

14


Therefore, 0 −

n 2|e(ΓG )| n = 2 − 1 and 2 − |v(ΓG )|

2|e(ΓG )| |v(ΓG )|

= 1. Hence, by (1.2) we have

LE(ΓG ) =(2n − 1)(2n − 1) + (22n − 2n+1 + 1)1 =2(2n − 1)2 .

By [15, Proposition 2.13], we also have n

Q-Spec(ΓG ) = {(2n+1 − 2)2 n G )| n Therefore, 2n+1 − 2 − 2|e(Γ |v(ΓG )| = 2 − 1 and 2 − 2 −

2n

−1

n+1

, (2n − 2)2 −2 +1 }. 2|e(ΓG )| |v(ΓG )| = 1. Therefore, by (1.3) we have

LE + (ΓG ) =(2n − 1)(2n − 1) + (22n − 2n+1 + 1)1 2

=2(2n − 1) .

Proposition 2.13. Let F = GF (pn ) where p is a prime. If G denotes the group     1 0 0   V (a, b, c) = a 1 0 : a, b, c ∈ F   b c 1

under matrix multiplication V (a, b, c)V (a′ , b′ , c′ ) = V (a + a′ , b + b′ + ca′ , c + c′ ), then E(ΓG ) = LE(ΓG ) = LE + (ΓG ) = 2(p3n − 2pn − 1). Proof. By [14, Proposition 5], we have 3n

Spec(ΓG ) = {(−1)p

−2pn −1

n

, (p2n − pn − 1)p

+1

}.

Therefore, by (1.1) we have E(ΓG ) = 2(p3n − 2pn − 1).

Note that |v(ΓG )| = pn (p2n − 1) and |e(ΓG )| =


p5n −p4n −2p3n +p2n +pn 2

since ΓG = (pn + 1)Kp2n −pn . Therefore,

2|e(ΓG )| = p2n − pn − 1. |v(ΓG )| 3n

n

n

L-Spec(ΓG ) = {0p +1 , (p2n − pn )p −2p −1 }. G )| G )| 2n Therefore, 0 − 2|e(Γ − pn − 1 and p2n − pn − 2|e(Γ |v(ΓG )| = p |v(ΓG )| = 1. Hence, by (1.2) we have LE(ΓG ) = (pn + 1)(p2n − pn − 1) + (p3n − 2pn − 1)1 = 2(p3n − 2pn − 1).

By [15, Proposition 2.14], we also have n

3n

n

Q-Spec(ΓG ) = {(2p2n − 2pn − 2)p +1 , (p2n − pn − 2)p −2p −1 }. 2n 2|e(ΓG )| G )| 2n n n = p − p − 1 and − p − 2 − Therefore, 2p2n − 2pn − 2 − 2|e(Γ p |v(ΓG )| |v(ΓG )| = 1. Therefore, by (1.3) we have LE + (ΓG ) = (pn + 1)(p2n − pn − 1) + (p3n − 2pn − 1)1 = 2(p3n − 2pn − 1).


15

3. Some consequences For a finite group G, the set CG (x) = {y ∈ G : xy = yx} is called the centralizer of an element x ∈ G. Let | Cent(G)| = |{CG (x) : x ∈ G}|, that is the number of distinct centralizers in G. A group G is called an n-centralizer group if | Cent(G)| = n. The study of these groups was initiated by Belcastro and Sherman [7] in the year 1994. The readers may conf. [11] for various results on these groups. In this section, we compute various energies of the commuting graphs of non-abelian n-centralizer finite groups for some positive integer n. We begin with the following result. Theorem 3.1. If G is a finite 4-centralizer group, then E(ΓG ) = LE(ΓG ) = LE + (ΓG ) = 6|Z(G)| − 6. Proof. Let G be a finite 4-centralizer group. Then, by [7, Theorem 2], we have by Theorem 2.2, the result follows.

G Z(G)

∼ = Z2 × Z2 . Therefore,

Further, we have the following result. Corollary 3.2. If G is a finite (p + 2)-centralizer p-group for any prime p, then E(ΓG ) = LE(ΓG ) = LE + (ΓG ) = 2(p2 − 1)|Z(G)| − 2(p + 1). Proof. Let G be a finite (p + 2)-centralizer p-group. Then, by [5, Lemma 2.7], we have Therefore, by Theorem 2.2, the result follows.

G Z(G)

∼ = Zp × Zp .

Theorem 3.3. If G is a finite 5-centralizer group, then E(ΓG ) = LE(ΓG ) = LE + (ΓG ) = 16|Z(G)| − 8.

or

E(ΓG ) = 10|Z(G)| − 8,

LE(ΓG ) =

and LE + (ΓG ) =

(

( 56|Z(G)|−40

, 5 12|Z(G)|2 +11|Z(G)|−10 , 5

16 5 , 12|Z(G)|2 +18|Z(G)|−30 , 5

if m = 3 and Z(G) = 1,2; otherwise

if m = 3 and Z(G) = 1; otherwise.

Proof. Let G be a finite 5-centralizer group. Then by [7, Theorem 4], we have G ∼ if Z(G) = Z3 × Z3 , then by Theorem 2.2, we have If

G Z(G)

G Z(G)

∼ = Z3 × Z3 or D6 . Now,

E(ΓG ) = LE(ΓG ) = LE + (ΓG ) = 16|Z(G)| − 8. ∼ = D6 , then by Theorem 2.4 we have ( 56|Z(G)|−40 , if m = 3 and Z(G) = 1,2; 5 E(ΓG ) = 10|Z(G)| − 8, LE(ΓG ) = 12|Z(G)| 2 +11|Z(G)|−10 , otherwise 5

and LE + (ΓG ) =

(

16 5 , 12|Z(G)|2 +18|Z(G)|−30 , 5

if m = 3 and Z(G) = 1; otherwise.


Let G be a finite group. The commutativity degree of G is given by the ratio Pr(G) =

|{(x, y) ∈ G × G : xy = yx}| . |G|2

The origin of the commutativity degree of a finite group lies in a paper of Erd¨ os and Tur´an (see [16]). Readers may conf. [8, 9, 23] for various results on Pr(G). In the following few results we shall compute

16


various energies of the commuting graphs of finite non-abelian groups G such that Pr(G) = r for some rational number r. Theorem 3.4. Let G be a finite group and p the smallest prime divisor of |G|. If Pr(G) = 2

p2 +p−1 , p3

then

E(ΓG ) = LE(ΓG ) = LE + (ΓG ) = 2(p2 − 1)|Z(G)| − 2(p + 1).

, then by [20, Theorem 3], we have Proof. If Pr(G) = p +p−1 p3 follows from Theorem 2.2.

G Z(G)

is isomorphic to Zp × Zp . Hence the result

As a corollary we have Corollary 3.5. Let G be a finite group such that Pr(G) = 58 . Then E(ΓG ) = LE(ΓG ) = LE + (ΓG ) = 6|Z(G)| − 6.

5 2 11 1 7 16 Theorem 3.6. If Pr(G) ∈ { 14 , 5 , 27 , 2 }, then E(ΓG ) ∈ {11, 7, 6, 3}, LE(ΓG ) ∈ { 480 13 , 16, 3 , 5 } and 16 LE + (ΓG ) ∈ { 370 13 , 6, 5 }. 5 2 11 1 Proof. If Pr(G) ∈ { 14 , 5 , 27 , 2 }, then as shown in [26, pp. 246] and [24, pp. 451], we have to one of the groups in {D14 , D10 , D8 , D6 }. Hence the result follows from Corollary 2.6.

G Z(G)

is isomorphic

Recall that genus of a graph is the smallest non-negative integer n such that the graph can be embedded on the surface obtained by attaching n handles to a sphere. A graph is said to be planar or toroidal if the genus of the graph is zero or one respectively. In the next two results we compute various energies of ΓG if ΓG is planar or toroidal. We begin with the following lemma. Lemma 3.7. Let G be a group isomorphic to any of the following groups (1) Z2 × D8 (2) Z2 × Q8 (3) M16 = ha, b : a8 = b2 = 1, bab = a5 i (4) Z4 ⋊ Z4 = ha, b : a4 = b4 = 1, bab−1 = a−1 i (5) D8 ∗ Z4 = ha, b, c : a4 = b2 = c2 = 1, ab = ba, ac = ca, bc = a2 cbi (6) SG(16, 3) = ha, b : a4 = b4 = 1, ab = b−1 a−1 , ab−1 = ba−1 i. Then E(ΓG ) = LE(ΓG ) = LE + (ΓG ) = 18. Proof. If G is isomorphic to any of the above listed groups, then |G| = 16 and |Z(G)| = 4. Therefore, G ∼ Z(G) = Z2 × Z2 . Thus the result follows from Theorem 2.2. Theorem 3.8. Let ΓG be the commuting graph of a finite non-abelian group G. If ΓG is planar then √ √ E(ΓG ) ∈{3, 6, 7, 12, 18, 26, 30, 76, 17 + 4 5 + 17}, √ 16 7 72 140 504 408 3924 526 + 46 13 LE(ΓG ) ∈{ , , 16, 18, , 6, , , , , } and 5 3 5 11 19 11 59 23 54 256 484 312 3844 756 16 , , , , }. LE + (ΓG ) ∈{ , 6, , 18, 5 5 11 19 11 59 23 Proof. By [2, Theorem 2.2], we have that ΓG is planar if and only if G is isomorphic to either D6 , D8 , D10 , D12 , Q8 , Q12 , Z2 × D8 , Z2 × Q8 , M16 , Z4 ⋊ Z4 , D8 ∗ Z4 , SG(16, 3), A4 , A5 , S4 , SL(2, 3) or Sz(2). If G ∼ = D6 , D8 , D10 or D12 , then by Corollary 2.6, we have 16 7 72 16 54 E(ΓG ) ∈ {3, 6, 7, 12}, LE(ΓG ) ∈ { , , 7, } and LE + (ΓG ) ∈ { , 6, }. 5 3 5 5 5 If G ∼ = Q8 or Q12 then, by Corollary 2.7, we have 54 72 and LE + (ΓG ) = 6 or . E(ΓG ) = 6 or 12, LE(ΓG ) = 6 or 5 5


17

If G ∼ = Z2 × D8 , Z2 × Q8 , M16 , Z4 ⋊ Z4 , D8 ∗ Z4 or SG(16, 3), then by Lemma 3.7, we have E(ΓG ) = LE(ΓG ) = LE + (ΓG ) = 18.

If G ∼ = A4 = ha, b : a2 = b3 = (ab)3 = 1i, then it can be seen that ΓG = K3 ⊔ 4K2 and so 140 256 E(ΓG ) = 12, LE(ΓG ) = and LE + (ΓG ) = . 11 11 If G ∼ = Sz(2), then by Theorem 2.1, we have 484 504 and LE + (ΓG ) = . E(ΓG ) = 26, LE(ΓG ) = 19 19 If G is isomorphic to SL(2, 3), then it was shown in the proof of [14, Theorem 4] and [15, Theorem 5.2] that Therefore

Spec(ΓG ) = {(−1)15 , 13 , 34 }, L-Spec(ΓG ) = {07 , 23 , 412 } and Q-Spec(ΓG ) = {03 , 215 , 64 }. E(ΓG ) = 30,

If G ∼ = A5 , then by Proposition 2.10, we have E(ΓG ) = 76,

408 11

and

LE + (ΓG ) =

312 . 11

3924 59

and

LE + (ΓG ) =

3844 . 59

LE(ΓG ) =

LE(ΓG ) =

noting that P SL(2, 4) ∼ = A5 . Finally, if G ∼ = S4 , then it was shown in [14, 15] that   √ !1 √ !1   √ √ 3 + 3 − 17 17 , . Spec(ΓG ) = 17 , (−1)10 , ( 5)2 , (− 5)2 ,   2 2 n o √ √ L-Spec(ΓG ) = 05 , 13 , 24 , 36 , 51 , (4 + 13)2 , (4 − 13)2 .

and

Q-Spec(ΓG ) = Therefore,

  

√ √ 04 , 16 , 24 , 33 , 51 , (4 + 5)2 , (4 − 5)2 ,

√ √ E(ΓG ) = 17 + 4 5 + 17,

√ !1 11 + 41 , 2

√ 526 + 46 13 LE(ΓG ) = 23

and

 √ !1 11 − 41  .  2

LE + (ΓG ) =

756 . 23


Theorem 3.9. Let ΓG be the commuting graph of a finite non-abelian group G. If ΓG is toroidal, then E(ΓG ) ∈{11, 18, 42, 25, 22, 34}, 480 230 962 236 103 390 LE(ΓG ) ∈{ , , , , , 48, } and 13 7 15 7 4 11 370 192 480 677 408 LE + (ΓG ) ∈{ , , 185, , , 59, } 13 7 7 20 11 Proof. By Theorem 6.6 of [10], we have ΓG is toroidal if and only if G is isomorphic to either D14 , D16 , Q16 , QD16 , D6 × Z3 , A4 × Z2 or Z7 ⋊ Z3 . If G ∼ = D14 or D16 then, by Corollary 2.6, we have 230 370 192 480 or and LE + (ΓG ) = or . E(ΓG ) = 11 or 18, LE(ΓG ) = 13 7 13 7 If G ∼ = Q16 then, by Corollary 2.7, we have 962 E(ΓG ) = 42, LE(ΓG ) = and LE + (ΓG ) = 185. 15 If G ∼ = QD16 then, by Proposition 2.9, we have 480 236 and LE + (ΓG ) = . E(ΓG ) = 18, LE(ΓG ) = 7 7

18


If G ∼ = Z7 ⋊ Z3 then, by Proposition 2.8, we have E(ΓG ) = 25, If G is isomorphic to D6 × Z3 , then

E(ΓG ) = 22,

103 4

and

LE + (ΓG ) =

LE(ΓG ) = 48

and

LE + (ΓG ) = 59.

390 11

and

LE + (ΓG ) =

LE(ΓG ) =

Finaly, if A4 × Z2 , then E(ΓG ) = 34,

LE(ΓG ) =

677 . 20

408 . 11


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