Common neighborhood spectrum of commuting graphs of finite groups
The commuting graph of a finite non-abelian group G with center Z(G), denoted by Гc(G), 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. In this paper, we compute the common neighborhood spectrum of commuting graphs of s...
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| Цитувати: | Common neighborhood spectrum of commuting graphs of finite groups / W.N.T. Fasfous, R. Sharafdini, R.K. Nath // Algebra and Discrete Mathematics. — 2021. — Vol. 32, № 1. — С. 33–48. — Бібліогр.: 24 назв. — англ. |
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irk-123456789-1887162023-03-13T19:10:46Z Common neighborhood spectrum of commuting graphs of finite groups Fasfous, W.N.T. Sharafdini, R. Nath, R.K. The commuting graph of a finite non-abelian group G with center Z(G), denoted by Гc(G), 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. In this paper, we compute the common neighborhood spectrum of commuting graphs of several classes of finite non-abelian groups and conclude that these graphs are CN-integral. 2021 Article Common neighborhood spectrum of commuting graphs of finite groups / W.N.T. Fasfous, R. Sharafdini, R.K. Nath // Algebra and Discrete Mathematics. — 2021. — Vol. 32, № 1. — С. 33–48. — Бібліогр.: 24 назв. — англ. 1726-3255 DOI:10.12958/adm1332 2020 MSC: 20D99, 05C50, 15A18, 05C25 http://dspace.nbuv.gov.ua/handle/123456789/188716 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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The commuting graph of a finite non-abelian group G with center Z(G), denoted by Гc(G), 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. In this paper, we compute the common neighborhood spectrum of commuting graphs of several classes of finite non-abelian groups and conclude that these graphs are CN-integral. |
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Fasfous, W.N.T. Sharafdini, R. Nath, R.K. |
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Fasfous, W.N.T. Sharafdini, R. Nath, R.K. Common neighborhood spectrum of commuting graphs of finite groups Algebra and Discrete Mathematics |
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Fasfous, W.N.T. Sharafdini, R. Nath, R.K. |
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Fasfous, W.N.T. |
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Common neighborhood spectrum of commuting graphs of finite groups |
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Common neighborhood spectrum of commuting graphs of finite groups |
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Common neighborhood spectrum of commuting graphs of finite groups |
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Common neighborhood spectrum of commuting graphs of finite groups |
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Common neighborhood spectrum of commuting graphs of finite groups |
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common neighborhood spectrum of commuting graphs of finite groups |
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Інститут прикладної математики і механіки НАН України |
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Common neighborhood spectrum of commuting graphs of finite groups / W.N.T. Fasfous, R. Sharafdini, R.K. Nath // Algebra and Discrete Mathematics. — 2021. — Vol. 32, № 1. — С. 33–48. — Бібліогр.: 24 назв. — англ. |
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Algebra and Discrete Mathematics |
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2025-07-16T10:54:16Z |
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1837800627961331712 |
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“adm-n3” — 2021/11/8 — 20:27 — page 33 — #35
© Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 32 (2021). Number 1, pp. 33–48
DOI:10.12958/adm1332
Common neighborhood spectrum of commuting
graphs of finite groups
W. N. T. Fasfous, R. Sharafdini, and R. K. Nath∗
Communicated by I. Ya. Subbotin
Abstract. The commuting graph of a finite non-abelian
group G with center Z(G), denoted by Γc(G), 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. In this paper, we compute the
common neighborhood spectrum of commuting graphs of several
classes of finite non-abelian groups and conclude that these graphs
are CN-integral.
1. Introduction
Let G be a simple graph whose vertex set is V (G) = {v1, v2, . . . , vn}.
The common neighborhood of two distinct vertices vi and vj , denoted
by C(vi, vj), is the set of vertices adjacent to both vi and vj other than
vi and vj . The common neighborhood matrix of G, denoted by CN(G),
is a matrix of size n whose (i, j)th entry is 0 or |C(vi, vj)| according
as i = j or i 6= j. Alwardi et al. have introduced and studied this
matrix in [4]. The set of all the eigenvalues of CN(G) with multiplicities
denoted by CN-spec(G) is called the common neighborhood spectrum, in
short CN-spectrum, of G. If α1, α2, . . . , αk are the eigenvalues of CN(G)
with multiplicities a1, a2, . . . , ak respectively then we write CN-spec(G) =
{αa1
1 , αa2
2 , . . . , α
ak
k }. A graph G is called CN-integral if CN-spec(G) contains
only integers.
∗Corresponding author.
2020 MSC: 20D99, 05C50, 15A18, 05C25.
Key words and phrases: commuting graph, spectrum, integral graph, finite
group.
https://doi.org/10.12958/adm1332
“adm-n3” — 2021/11/8 — 20:27 — page 34 — #36
34 Common neighborhood spectrum of graphs
The commuting graph of a finite non-abelian group G with center
Z(G) is a simple undirected graph whose vertex set is G \ Z(G) and two
vertices x and y are adjacent if and only if xy = yx. We write Γc(G) to
denote this graph. In [5, 12–14, 16, 18, 21, 23], various aspects of Γc(G) are
studied. In section 2 of this paper, we derive a computing formula for
CN-spectrum of a particular class of graphs and list a few useful results.
In section 3, we compute CN-spectrum of commuting graph of groups G
such that G
Z(G) is isomorphic to the Suzuki group of order 20, Zp × Zp
(where p is a prime) and a dihedral group of order 2m. In section 4,
we compute CN-spectrum of commuting graphs of several well-known
groups including the quasidihedral groups, projective special linear groups,
general linear groups etc. As consequences of our results, in section 5, we
show that commuting graphs of all the groups considered in section 3
and section 4 are CN-integral. We shall determine some positive integers
n such that Γc(G) is CN-integral if G is an n-centralizer group. Recall
that a group G is called an n-centralizer group if |Cent(G)| = n, where
Cent(G) = {CG(x) : x ∈ G} and CG(x) = {y ∈ G : xy = yx} is
the centralizer of x. The study of n-centralizer groups was initiated by
Belcastro and Sherman [7] in 1994. The reader may conf. [11] for various
results on n-centralizer groups. We shall also determine some positive
rational numbers r such that Γc(G) is CN-integral if the commutativity
degree of G is r. Recall that the commutativity degree of G, denoted by
Pr(G), is the probability that a randomly chosen pair of elements of G
commute. The origin of commutativity degree of a finite group lies in a
paper of Erdös and Turán (see [15]). The reader may conf. [8, 9, 19, 22]
for various results regarding this notion. Further, we show that Γc(G) is
CN-integral if Γc(G) is planar or toroidal and G is not isomorphic to S4,
the symmetric group of degree 4. Note that a graph is planar or toroidal
according as its genus is zero or one respectively. Also, the 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.
It is worth mentioning that Afkhami et al. [3] and Das et al. [10] have
classified all finite non-abelian groups whose commuting graphs are planar
or toroidal recently.
2. A useful formula and prerequisites
We write G = G1 ⊔G2 to denote that G has two components namely G1
and G2. Also, lKm denotes the disjoint union of l copies of the complete
graph Km on m vertices. We begin this section with the following lemma.
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W. N. T. Fasfous, R. Sharafdini, R. K. Nath 35
Lemma 1. If G = G1 ⊔G2 ⊔ · · · ⊔ Gm then CN-spec(G) =
k
∪
i=1
CN-spec(Gi)
counting the multiplicities.
Lemma 2. If Kn denotes the complete graph on n vertices then
CN-spec(Kn) = {(−(n− 2))n−1, ((n− 1)(n− 2))1}.
Proof. Let A(Kn) be the adjacency matrix of Kn. Then we have CN(Kn)
= (n− 2)A(Kn). Hence, the result follows.
Now we derive a formula for CN-spectrum of graphs that are disjoint
union of some complete graphs. The following theorem is very useful in
order to compute CN-spectrum of commuting graphs of some classes of
finite groups.
Theorem 1. Let G = l1Km1 ⊔ l2Km2 ⊔ · · · ⊔ lkKmk
, where liKmi
denotes
disjoint union of li copies of the complete graphs Kmi
on mi vertices for
1 6 i 6 k. Then
CN-spec(G) = {(−(m1−2))l1(m1−1), ((m1 − 1)(m1 − 2))l1 , . . . ,
(−(mk − 2))lk(mk−1), ((mk − 1)(mk − 2))lk}.
Proof. Let G = G1⊔G2⊔· · ·⊔Gk. Then, by Lemma 1, we have CN-spec(G)
=
k
∪
i=1
CN-spec(Gi) counting the multiplicities. Therefore, using Lemma 2,
we have
CN-spec(liKmi
) = {(−(mi − 2))li(mi−1), ((mi − 1)(mi − 2))li}.
Hence, the result follows by considering Gi = liKmi
for 1 6 i 6 k.
We conclude this section with the following useful results.
Theorem 2. [7, Theorem 2] If G is a finite 4-centralizer group then
G
Z(G)
∼= Z2 × Z2.
Theorem 3. [6, Lemma 2.7] If G is a finite (p+ 2)-centralizer p-group
then G
Z(G)
∼= Zp × Zp.
Theorem 4. [7, Theorem 4] If G is a finite 5-centralizer group then
G
Z(G)
∼= Z3 × Z3 or D6.
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36 Common neighborhood spectrum of graphs
Theorem 5. [2, Lemma 2.4] Let G be a finite non-abelian group and
{x1, x2, . . . , xr} be a set of pairwise non-commuting elements of G having
maximal size. Then G is a 4-centralizer or a 5-centralizer group according
as r = 3 or 4.
Theorem 6. [17, Theorem 3] Let G be a finite group and p the smallest
prime divisor of |G|. Then Pr(G) = p2+p−1
p3
if and only if G
Z(G)
∼= Zp × Zp.
Theorem 7. [8, Proposition 3.3.7] If G is a finite non-solvable group with
Pr(G) = 1
12 then G ∼= A5 ×B for some finite abelian group B.
Theorem 8. [3, Theorem 2.2] Let G be a finite non-abelian group. Then
Γc(G) 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).
Theorem 9. [10, Theorem 6.6] Let G be a finite non-abelian group.
Then Γc(G) is toroidal if and only if G is isomorphic to either D14, D16,
Q16, QD16, D6 × Z3, A4 × Z2 or Z7 ⋊ Z3.
Theorem 10. [1, Proposition 2. 3] Let G be a finite non-abelian group.
Then the complement of Γc(G) is planar if and only if G is isomorphic to
either D6, D8 or Q8.
3. Groups having known central quotient
In this section, we compute CN-spectrum of commuting graphs of
finite non-abelian groups having well-known central quotient such as the
Suzuki group of order 20, Zp × Zp (where p is a prime) and the dihedral
groups. We begin with the following lemma from [12] and [13].
Lemma 3. Let G be a finite group with center Z(G). If G
Z(G) is isomor-
phic to
(a) the Suzuki group Sz(2), presented by 〈a, b : a5 = b4 = 1, b−1ab = a2〉,
then Γc(G) = K4|Z(G)| ⊔ 5K3|Z(G)|.
(b) Zp × Zp, where p is a prime, then Γc(G) = (p+ 1)K(p−1)|Z(G)|.
(c) the dihedral group D2m (m > 2), presented by 〈a, b : am = b2 = 1,
bab−1 = a−1〉, then Γc(G) = K(m−1)|Z(G)| ⊔mK|Z(G)|.
Now we have the following main result of this section.
Theorem 11. Let G be a finite group with center Z(G). If G
Z(G) is iso-
morphic to
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W. N. T. Fasfous, R. Sharafdini, R. K. Nath 37
(a) the Suzuki group Sz(2), presented by 〈a, b : a5 = b4 = 1, b−1ab = a2〉,
then CN-spec(Γc(G)) is given by
{(−(4|Z(G)− 2))4|Z(G)−1, ((4|Z(G)− 1)(4|Z(G)− 2))1,
(−(3|Z(G)| − 2))5(3|Z(G)|−1), ((3|Z(G)| − 1)(3|Z(G)| − 2))5}.
(b) Zp × Zp, where p is a prime, then CN-spec(Γc(G)) is given by
{(−((p− 1)|Z(G)|−2))(p+1)((p−1)|Z(G)|−1),
(((p− 1)|Z(G)| − 1)((p− 1)|Z(G)| − 2))p+1}.
(c) the dihedral group D2m (m > 2), presented by 〈a, b : am = b2 =
1, bab−1 = a−1〉, then CN-spec(Γc(G)) is given by
{(−((m− 1)|Z(G)| − 2))(m−1)|Z(G)|−1,
(((m− 1)|Z(G)| − 1)((m− 1)|Z(G)| − 2))1,
(−(|Z(G)| − 2))m(|Z(G)|−1), ((|Z(G)| − 1)(|Z(G)| − 2))m}.
Proof. (a) If G
Z(G)
∼= Sz(2) then, by Lemma 3(a), we have Γc(G) =
K4|Z(G)| ⊔ 5K3|Z(G)|. Therefore, by Theorem 1, we have
CN-spec(Γc(G)) =
{(−(4|Z(G)− 2))4|Z(G)−1, ((4|Z(G)− 1)(4|Z(G)− 2))1,
(−(3|Z(G)| − 2))5(3|Z(G)|−1), ((3|Z(G)| − 1)(3|Z(G)| − 2))5}.
(b) If G
Z(G)
∼= Zp × Zp then, by Lemma 3(b), we have Γc(G) =
(p+ 1)K(p−1)|Z(G)|. Therefore, by Theorem 1, we have
CN-spec(Γc(G)) = {(−((p− 1)|Z(G)| − 2))(p+1)((p−1)|Z(G)|−1),
(((p− 1)|Z(G)| − 1)((p− 1)|Z(G)| − 2))p+1}.
(c) If G
Z(G)
∼= D2m then, by Lemma 3(c), we have
Γc(G) = K(m−1)|Z(G)| ⊔mK|Z(G)|.
Therefore, by Theorem 1,
CN-spec(Γc(G)) = {(−((m− 1)|Z(G)| − 2))(m−1)|Z(G)|−1,
(((m− 1)|Z(G)| − 1)((m− 1)|Z(G)| − 2))1,
(−(|Z(G)| − 2))m(|Z(G)|−1), ((|Z(G)| − 1)(|Z(G)| − 2))m}.
This completes the proof.
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38 Common neighborhood spectrum of graphs
We conclude this section with the following corollaries of Theorem 11.
Corollary 1. Let G be a group isomorphic to one of the following groups
(a) Z2 ×D8
(b) Z2 ×Q8
(c) M16 = 〈a, b : a8 = b2 = 1, bab = a5〉
(d) Z4 ⋊ Z4 = 〈a, b : a4 = b4 = 1, bab−1 = a−1〉
(e) D8 ∗ Z4 = 〈a, b, c : a4 = b2 = c2 = 1, ab = ba, ac = ca, bc = a2cb〉
(f) SG(16, 3) = 〈a, b : a4 = b4 = 1, ab = b−1a−1, ab−1 = ba−1〉.
Then CN-spec(Γc(G)) = {(−2)9, 63}.
Proof. We have |G| = 16 and |Z(G)| = 4. Therefore, G
Z(G)
∼= Z2 × Z2.
Hence, putting p = 2 and |Z(G)| = 4 in Theorem 11(b) we get the required
result.
Corollary 2. Let G be a non-abelian group.
(a) If G is of order p3, for any prime p, then
CN-spec(Γc(G)) = {(−(p2 − p−2))(p+1)(p2−p−1),
((p2 − p− 1)(p2 − p− 2))p+1}.
(b) Let G be the metacyclic group M2mn (m > 3), presented by 〈a, b :
am = b2n = 1, bab−1 = a−1〉. If m is odd then CN-spec(Γc(M2mn)) is
given by
{(−(mn− n− 2))mn−n−1, ((mn− n− 1)(mn− n− 2))1,
(−(n− 2))mn−m, ((n− 1)(n− 2))m}.
If m is even then CN-spec(Γc(M2mn)) is given by
{(−(mn− 2n− 2))mn−2n−1, ((mn− 2n− 1)(mn− 2n− 2))1,
(−(2n− 2))
m(2n−1)
2 , ((2n− 1)(2n− 2))
m
2 }.
(c) If G is the dihedral group D2m (m > 3), presented by 〈a, b : am =
b2 = 1, bab−1 = a−1〉, then
CN-spec(Γc(G))
=
{
{(−(m− 3))m−2, ((m− 2)(m− 3))1, 0m}, if m is odd
{(−(m− 4))m−3, ((m− 3)(m− 4))1, 0m}, if m is even.
(d) If G is the generalized quaternion group Q4n (n > 2), presented by
〈x, y : y2n = 1, x2 = yn, xyx−1 = y−1〉, then
CN-spec(Γc(G)) = {(−(2n− 4))2n−3, ((2n− 3)(2n− 4))1, 02n}.
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W. N. T. Fasfous, R. Sharafdini, R. K. Nath 39
Proof. (a) If G is of order p3 then |Z(G)| = p and G
Z(G)
∼= Zp × Zp.
Therefore, putting |Z(G)| = p, in Theorem 11(b), we get
CN-spec(Γc(G)) =
{(−(p2 − p− 2))(p+1)(p2−p−1), ((p2 − p− 1)(p2 − p− 2))p+1}.
(b) If m is odd then |Z(M2mn)| = n and M2mn
Z(M2mn)
∼= D2m. Therefore,
putting |Z(G)| = n, in Theorem 11(c), we get
CN-spec(Γc(M2mn)) = {(−(mn− n− 2))mn−n−1,
((mn− n− 1)(mn− n− 2))1, (−(n− 2))mn−m, ((n− 1)(n− 2))m}.
If m is even then |Z(M2mn)| = 2n and M2mn
Z(M2mn)
∼= Dm. Therefore, putting
|Z(G)| = 2n and replacing m by m
2 , in Theorem 11(c), we get
CN-spec(Γc(M2mn)) =
{(−(mn− 2n− 2))mn−2n−1, ((mn− 2n− 1)(mn− 2n− 2))1,
(−(2n− 2))
m(2n−1)
2 , ((2n− 1)(2n− 2))
m
2 }.
(c) Follows from part (b), considering n = 1.
(d) Note that |Z(Q4n)| = 2 and Q4n
Z(Q4n)
∼= D2n. Therefore, putting
|Z(G)| = 2 and m = n in Theorem 11(c), we get the required result.
4. More classes of groups
In this section, we compute CN-spectrum of commuting graphs of
several well-known groups including the quasidihedral groups, projective
special linear groups, general linear groups etc. We begin with the following
useful results from [12].
Lemma 4. Let G be a non-abelian group. If G is isomorphic to
(a) a group of order pq, where p and q are primes with p | (q − 1), then
Γc(G) = Kq−1 ⊔ qKp−1.
(b) the quasidihedral group QD2n (n > 4), presented by 〈a, b : a2n−1
=
b2 = 1, bab−1 = a2
n−2−1〉, then Γc(G) = K2n−1−2 ⊔ 2n−2K2.
(c) the projective special linear group PSL(2, 2k), where k > 2, then
Γc(G) = (2k + 1)K2k−1 ⊔ 2k−1(2k + 1)K2k−2 ⊔ 2k−1(2k − 1)K2k .
(d) the general linear group GL(2, q), where q = pn > 2 and p is a
prime, then
Γc(G) =
q(q + 1)
2
Kq2−3q+2 ⊔
q(q − 1)
2
Kq2−q ⊔ (q + 1)Kq2−2q+1.
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40 Common neighborhood spectrum of graphs
Lemma 5. Let G be a non-abelian group. If G is isomorphic to
(a) the Hanaki group A(n, ϑ) (n > 2) of order 22n given by
U(a, b) =
1 0 0
a 1 0
b ϑ(a) 1
: a, b ∈ F
under matrix multiplication U(a, b)U(a′, b′) := U(a + a′, b + b′ +
a′ϑ(a)), where F = GF (2n) and ϑ be the Frobenius automorphism
of F given by ϑ(x) = x2 for all x ∈ F , then Γc(G) = (2n − 1)K2n .
(b) the Hanaki group A(n, p) of order p3n given by
V (a, b, c) =
1 0 0
a 1 0
b c 1
: a, b, c ∈ F
under matrix multiplication V (a, b, c)V (a′, b′, c′) := V (a+ a′, b+ b′+
ca′, c + c′), where F = GF (pn) and p is a prime, then Γc(G) =
(pn + 1)Kp2n−pn .
Now, we compute CN-spec(Γc(G)) for more families of finite groups.
Theorem 12. Let G be a non-abelian group.
(a) If G is of order pq, where p and q are primes with p | (q − 1), then
CN-spec(Γc(G)) is given by
{(−(q− 3))q−2, ((q− 2)(q− 3))1, (−(p− 3))pq−2q, ((p− 2)(p− 3))q}.
(b) If G is the quasidihedral group QD2n (n > 4), presented by 〈a, b :
a2
n−1
= b2 = 1, bab−1 = a2
n−2−1〉, then CN-spec(Γc(G)) is given by
{(−(2n−1 − 4))2
n−1−3, ((2n−1 − 3)(2n−1 − 4))1, 02
n−1}.
(c) If G is the projective special linear group PSL(2, 2k), where k > 2,
then CN-spec(Γc(G)) is given by
{(−(2k − 3))(2
k+1)(2k−2), ((2k − 2)(2k − 3))2
k+1,
(−(2k − 4))2
k−1(2k+1)(2k−3), ((2k − 3)(2k − 4))2
k−1(2k+1),
(−(2k − 2))2
k−1(2k−1)2 , ((2k − 1)(2k − 2))2
k−1(2k−1)}.
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W. N. T. Fasfous, R. Sharafdini, R. K. Nath 41
(d) If G is the general linear group GL(2, q), where q = pn > 2 and p is
a prime, then CN-spec(Γc(G)) is given by
{(−(q2 − 3q))
q(q+1)(q2−3q+1)
2 , ((q2 − 3q + 1)(q2 − 3q))
q(q+1)
2 ,
(−(q2 − q − 2))
q(q−1)(q2−q−1)
2 , ((q2 − q − 1)(q2 − q − 2))
q(q−1)
2 ,
(−(q2 − 2q − 1))(q+1)(q2+2q), ((q2 − 2q)(q2 − 2q − 1))q+1}.
Proof. (a) By Lemma 4(a), we have Γc(G) = Kq−1 ⊔ qKp−1. Therefore,
by Theorem 1, we have
CN-spec(Γc(G)) =
{(−(q − 3))q−2, ((q − 2)(q − 3))1, (−(p− 3))pq−2q, ((p− 2)(p− 3))q}.
(b) By Lemma 4(b), we have Γc(QD2n) = K2n−1−2 ⊔ 2n−2K2. There-
fore, by Theorem 1, we have
CN-spec(Γc(QD2n)) =
{(−(2n−1 − 4))2
n−1−3, ((2n−1 − 3)(2n−1 − 4))1, 02
n−1}.
(c) By Lemma 4(c), we have
Γc(G) = (2k + 1)K2k−1 ⊔ 2k−1(2k + 1)K2k−2 ⊔ 2k−1(2k − 1)K2k .
Therefore, by Theorem 1, we have
CN-spec(Γc(G)) = {(−(2k − 3))(2
k+1)(2k−2), ((2k − 2)(2k − 3))2
k+1,
(−(2k − 4))2
k−1(2k+1)(2k−3), ((2k − 3)(2k − 4))2
k−1(2k+1),
(−(2k − 2))2
k−1(2k−1)2 , ((2k − 1)(2k − 2))2
k−1(2k−1)}.
(d) By Lemma 4(d), we have
Γc(G) =
q(q + 1)
2
Kq2−3q+2 ⊔
q(q − 1)
2
Kq2−q ⊔ (q + 1)Kq2−2q+1.
Therefore, by Theorem 1, we have
CN-spec(Γc(G)) =
{(−(q2 − 3q))
q(q+1)(q2−3q+1)
2 , ((q2 − 3q + 1)(q2 − 3q))
q(q+1)
2 ,
(−(q2 − q − 2))
q(q−1)(q2−q−1)
2 , ((q2 − q − 1)(q2 − q − 2))
q(q−1)
2 ,
(−(q2 − 2q − 1))(q+1)(q2+2q), ((q2 − 2q)(q2 − 2q − 1))q+1}.
This completes the proof.
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42 Common neighborhood spectrum of graphs
Theorem 13. Let G be a non-abelian group.
(a) If G is the Hanaki group A(n, ϑ) (n > 2) of order 22n given by
U(a, b) =
1 0 0
a 1 0
b ϑ(a) 1
: a, b ∈ F
under matrix multiplication U(a, b)U(a′, b′) := U(a + a′, b + b′ +
a′ϑ(a)), where F = GF (2n) and ϑ is the Frobenius automorphism
of F given by ϑ(x) = x2 ∀x ∈ F , then CN-spec(Γc(G)) is given by
{(−(2n − 2))(2
n−1)2 , ((2n − 1)(2n − 2))2
n−1}.
(b) If G is the Hanaki group A(n, p) of order p3n given by
V (a, b, c) =
1 0 0
a 1 0
b c 1
: a, b, c ∈ F
under matrix multiplication V (a, b, c)V (a′, b′, c′) := V (a+ a′, b+ b′+
ca′, c+ c′), where F = GF (pn) and p is a prime, then
CN-spec(Γc(G)) = {(−(p2n−pn − 2))(p
n+1)(p2n−pn−1),
((p2n − pn − 1)(p2n − pn − 2))p
n+1}.
Proof. (a) By Lemma 5(a), we have Γc(A(n, ϑ)) = (2n−1)K2n . Therefore,
by Theorem 1, we have
CN-spec(Γc(A(n, ϑ))) = {(−(2n − 2))(2
n−1)2 , ((2n − 1)(2n − 2))2
n−1}.
(b) By Lemma 5(b), we have Γc(A(n, p)) = (pn+1)Kp2n−pn . Therefore,
by Theorem 1, we have
CN-spec(Γc(A(n, p))) = {(−(p2n−pn − 2))(p
n+1)(p2n−pn−1),
((p2n − pn − 1)(p2n − pn − 2))p
n+1}.
This completes the proof.
Note that all the groups considered above are abelian centralizer group
(in short, AC-group). In other words, CG(x) is abelian for all x ∈ G\Z(G).
In the following two results we compute CN-spectrum of commuting
graphs of finite AC-groups.
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W. N. T. Fasfous, R. Sharafdini, R. K. Nath 43
Theorem 14. Let G be a finite non-abelian AC-group with distinct cen-
tralizers X1, . . . , Xn of non-central elements of G. Then CN-spec(Γc(G))
is given by the set
{(−(|X1| − |Z(G)| − 2))|X1|−|Z(G)|−1,
((|X1| − |Z(G)| − 1)(|X1| − |Z(G)| − 2))1, . . . ,
(−(|Xn| − |Z(G)| − 2))|Xn|−|Z(G)|−1,
((|Xn| − |Z(G)| − 1)(|Xn| − |Z(G)| − 2))1}.
Proof. By [12, Lemma 1], we have Γc(G) =
n
⊔
i=1
K|Xi|−|Z(G)|. Therefore,
the result follows from Theorem 1.
Corollary 3. Let G ∼= H ×A where H is a finite non-abelian AC-group
and A is any finite abelian group. Then CN-spec(Γc(H ×A)) is given by
the set
{(−((|X1| − |Z(H)|)|A| − 2))(|X1|−|Z(H)|)|A|−1,
(((|X1| − |Z(H)|)|A| − 1)((|X1| − |Z(H)|)|A| − 2))1, . . . ,
(−((|Xn| − |Z(H)|)|A| − 2))(|Xn|−|Z(H)|)|A|−1,
(((|Xn| − |Z(H)|)|A| − 1)((|Xn| − |Z(H)|)|A| − 2))1},
where X1, . . . , Xn are the distinct centralizers of non-central elements
of H.
Proof. Let H be a finite non-abelian AC-group and A be any finite abelian
group then Z(H×A) = Z(H)×A. Further, if X1, . . . , Xn are the distinct
centralizers of non-central elements of H then the distinct centralizers of
non-central elements of H ×A are given by X1 ×A,X2 ×A, . . . ,Xn ×A.
Therefore, H × A is also an AC-group. Hence, the result follows from
Theorem 14.
5. Consequences
In this section, we record some consequences of the results obtained
in earlier sections. Firstly, note that CN-spectrum of commuting graphs
of all the groups considered in section 3 and section 4 contain only inte-
gers. Therefore, commuting graphs of those groups are CN-integral. The
following results show that the commuting graph of a finite n-centralizer
group is CN-integral if n = 4, 5.
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44 Common neighborhood spectrum of graphs
Theorem 15. If G is a finite 4-centralizer group then Γc(G) is CN-
integral.
Proof. Let G be a finite 4-centralizer group. Then, by Theorem 2, we
have G
Z(G)
∼= Z2 × Z2. Hence the result follows from Theorem 11(b) by
considering p = 2.
Further, we have the following result.
Theorem 16. Let G be a finite (p+2)-centralizer p-group for any prime p.
Then Γc(G) is CN-integral.
Proof. Let G be a finite (p+ 2)-centralizer p-group. Then, by Theorem 3,
we have G
Z(G)
∼= Zp×Zp. Hence the result follows from Theorem 11(b).
Theorem 17. If G is a finite 5-centralizer group then Γc(G) is CN-
integral.
Proof. Let G be a finite 5-centralizer group. Then by Theorem 4 we have
G
Z(G)
∼= Z3 × Z3 or D6. Hence the result follows from Theorem 11, parts
(b) and (c).
As a corollary to Theorem 15 and Theorem 17 we have the following
result.
Corollary 4. Let G be a finite non-abelian group and {x1, x2, . . . , xr} be
a set of pairwise non-commuting elements of G having maximal size. Then
Γc(G) is CN-integral if r = 3, 4.
Proof. By Theorem 5, we have that G is a 4-centralizer or a 5-centralizer
group. Hence the result follows from Theorem 15 and Theorem 17.
The following theorems give some rational numbers r such that Γc(G)
is CN-integral if Pr(G) = r, where Pr(G) is the commutativity degree of
a finite group G.
Theorem 18. If Pr(G) ∈ { 5
14 ,
2
5 ,
11
27 ,
1
2 ,
7
16 ,
5
8} then Γc(G) is CN-integral.
Proof. If Pr(G) ∈ { 5
14 ,
2
5 ,
11
27 ,
1
2 ,
7
16 ,
5
8} then as shown in [24, pp. 246]
and [20, pp. 451], we have G
Z(G) is isomorphic to one of the groups in
{D14, D10, D8, D6,Z2 ×Z2,Z3 ×Z3}. Hence the result follows from Theo-
rem 11, parts (b) and (c).
Theorem 19. Let G be a finite group and p the smallest prime divisor
of |G|. If Pr(G) = p2+p−1
p3
then Γc(G) is CN-integral.
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W. N. T. Fasfous, R. Sharafdini, R. K. Nath 45
Proof. If Pr(G) = p2+p−1
p3
then, by Theorem 6, we have G
Z(G) is isomorphic
to Zp × Zp. Hence the result follows from Theorem 11(b).
Theorem 20. If G is a finite non-solvable group with Pr(G) = 1
12 then
Γc(G) is CN-integral.
Proof. By Theorem 7, we have that G is isomorphic to A5 ×B for some
finite abelian group B. Since A5 is an AC-group, the result follows from
Corollary 3.
The following three theorems show that Γc(G) is CN-integral if Γc(G)
is planar and G is not isomorphic to S4, troidal or the complement of
Γc(G) is planar.
Theorem 21. Let G be a finite non-abelian group. If Γc(G) is planar and
G is not isomorphic to S4 then Γc(G) is CN-integral.
Proof. By Theorem 8, 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, D12, Q8 or Q12 then, by Corollary 2 parts (c) and
(d), we have that Γc(G) is CN-integral. If G ∼= Z2×D8,Z2×Q8,M16,Z4⋊
Z4, D8 ∗ Z4 or SG(16, 3) then, by Corollary 1, it follows that Γc(G) is
CN-integral. If G ∼= A4 then it can be seen that Γc(G) = K3 ⊔ 4K2. Using
Theorem 1, we have CN-spec(Γc(G)) = {(−1)2, 21, 08}, hence Γc(G) in CN-
integral. If G ∼= Sz(2) then G
Z(G)
∼= Sz(2). Therefore, by Theorem 11(a),
it follows that Γc(G) is CN-integral. If G is isomorphic to SL(2, 3) then it
can be seen that Γc(G) = 3K2 ⊔ 4K4. Therefore, by Theorem 1, we have
CN-spec(Γc(G)) = {06, (−2)12, 64}, hence Γc(G) in CN-integral.
We have PSL(2, 4) ∼= A5. Therefore, if G ∼= A5 then by Theorem 12(c)
it follows that Γc(G) is CN-integral.
Finally, if G ∼= S4 then it can be seen that the characteristic polynomial
of CN(Γc(G)) is x8(x− 3)2(x+ 1)11(x2 − 5x− 30) and so
CN-spec(Γc(G)) =
08, 32, (−1)11,
(
5 +
√
145
2
)1
,
(
5−
√
145
2
)1
.
Hence, Γc(G) is not CN-integral. This completes the proof.
Theorem 22. Let G be a finite non-abelian group. If Γc(G) is toroidal
then Γc(G) is CN-integral.
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46 Common neighborhood spectrum of graphs
Proof. By Theorem 9, G is isomorphic to either D14, D16, Q16, QD16,
D6 × Z3, A4 × Z2 or Z7 ⋊ Z3.
If G ∼= D14, D16 or Q16 then, by Corollary 2 parts (c) and (d), it follows
that Γc(G) is CN-integral. If G ∼= QD16 then, by Theorem 12(b), we have
that Γc(G) is CN-integral. If G ∼= Z7 ⋊ Z3 then Γc(G) is CN-integral,
follows from Theorem 12(a) by considering p = 3 and q = 7. If G is
isomorphic to D6 × Z3 then G
Z(G)
∼= D6. Therefore, by Theorem 11(c),
Γc(G) is CN-integral. If G is isomorphic to A4 ×Z2 then by Corollary 3 it
follows that Γc(G) is CN-integral since A4 is an AC-group. This completes
the proof.
We also have the following result.
Theorem 23. Let G be a finite non-abelian group. If the complement of
Γc(G) is planar then Γc(G) is CN-integral.
Proof. By Theorem 10, G is isomorphic to either D6, D8 or Q8. Hence
the result follows from Corollary 2 parts (c) and (d).
In [12,13,21], Dutta and Nath have computed spectrum of the com-
muting graphs of all the groups considered in this paper. It was observed
that commuting graphs of all those groups except S4 are integral. The
commuting graph of S4 is neither integral nor CN-integral. Recall that a
graph is called integral if all the eigenvalues of its adjacency matrix are
integers. We conclude this paper with the following problems.
Problem 1. Let G be a finite non-abelian group. Does the fact “Γc(G)
is integral” imply Γc(G) is CN-integral?
More generally, one may pose the following problem.
Problem 2. Let G be any graph. Does the fact “G is integral” imply G is
CN-integral?
Acknowledgment
The authors would like to thank the referee for his/her valuable com-
ments and suggestions. The first author is thankful to Indian Council for
Cultural Relations for the ICCR Scholarship.
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W. N. T. Fasfous, R. Sharafdini, R. K. Nath 47
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Contact information
Walaa Nabil
Taha Fasfous,
Rajat Kanti Nath
Department of Mathematical Sciences, Tezpur
University, Napaam-784028, Sonitpur, Assam,
India
E-Mail(s): w.n.fasfous@gmail.com,
rajatkantinath@yahoo.com
Web-page(s): www.tezu.ernet.in/dmaths/
people/faculty-pages/rkn.
html
Reza Sharafdini Department of Mathematics, Faculty of Science,
Persian Gulf University, Bushehr 75169-13817,
Iran
E-Mail(s): sharafdini@pgu.ac.ir
Web-page(s): research.pgu.ac.ir/
~RSharafdini
Received by the editors: 09.02.2019.
mailto:w.n.fasfous@gmail.com
mailto:rajatkantinath@yahoo.com
www.tezu.ernet.in/dmaths/people/faculty-pages/rkn.html
www.tezu.ernet.in/dmaths/people/faculty-pages/rkn.html
www.tezu.ernet.in/dmaths/people/faculty-pages/rkn.html
mailto:sharafdini@pgu.ac.ir
research.pgu.ac.ir/~RSharafdini
research.pgu.ac.ir/~RSharafdini
W. N. T. Fasfous, R. Sharafdini, R. K. Nath
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