Finite cyclic group

4.2 Minimum degree

4.2.1 Finite cyclic group

In this subsection, We observe that δpGpZ_nqq is the degree of one of the proper divisors of n (cf. Lemma 4.2.3). We show that φpnq `1 is a sharp lower bound for δpGpZ_nqq. Further, we obtain some inequalities involving degrees of various elements of Z_n (cf. Proposition 4.2.6). Subsequently, we determineδpGpZ_nqqwhen n has two prime factors or n is a product of at most four distinct primes (cf. Theorem 4.2.7).

We conclude this subsection by giving two sharp upper bounds of δpGpZ_nqq.

Lemma 4.2.3. If n P N is a composite number, then there exists a proper divisor cą1 of n such that δpGpZnqq “degpcq.

Proof. Since n is a composite number, Z¹_n is non-empty. Note that degpaq ď n´1 for all aP Z_n. Moreover, by Remark 1.5.14, degpbq “n´1 for all b P SpZ_nq. Thus there exists a P Z¹_n such that δpGpZnqq “ degpaq. Now take c “ gcdpa, nq. Then cn, 1 ă c ă n and from Theorem 1.3.22, we have xcy “ xay. Hence by Lemma 4.2.1, δpGpZnqq “ degpcq.

66 Minimum Degree and Connectivity

Theorem 4.2.4. For any integer ną1,

(i) Ifnis a composite number, thenδpGpZ_nqq “φpnq`1`δpG¹pZ_nqq. Consequently, δpGpZ_nqq ěφpnq `1.

(ii) δpGpZnqq “φpnq `1 if and only if n “2p for some prime pě3.

Proof. (i) Let n PN be a composite number. In view of Lemma 4.2.3, it is enough to consider only elements of Z¹_n. Since each a P Z¹_n is adjacent to all elements of SpZ_nq and |SpZ_nq| “φpnq `1, we have deg<sub>GpZnq</sub>paq “ deg<sub>G</sub>1pZnqpaq `φpnq `1. Thus the proof follows.

(ii) Let n “2p for some prime pě3. Then by Proposition 2.3.3, G¹pZ_nq is discon- nected, and its component induced by xpy^˚ has p as its only vertex. Consequently, δpG¹pZ_nqq “0, and hence by (i), δpGpZ_nqq “φpnq `1.

Conversely, let δpGpZ_nqq “ φpnq `1. If n is a prime number, then δpGpZ<sub>n</sub>qq “ n´1 ‰ φpnq `1. So n is a composite number. Then by (i), δpG¹pZ_nqq “ 0, and hence G¹pZ_nq is disconnected. Accordingly, by Proposition 2.3.3, n is a product of two distinct primes; say n “ pq. It is easy to see that subgraphs induced by xpy^˚ and xqy^˚ are the only components of G¹pZ_nq. If both p and q are odd primes, then

|xpy^˚|,|xqy^˚| ě2 and hence δpG¹pZ_nqq ě1. As this is a contradiction, exactly one of p orq is 2. Hence takingq “2, we get n“2p.

The following lemma gives us a formula to compute degrees of vertices ofZ¹_n in GpZ_nq.

Lemma 4.2.5 ([Moghaddamfar et al., 2014]). Suppose the integer n ą 1 is not a prime power and aP Z¹_n. If b“gcdpa, nq, then in GpZ_nq,

degpaq “ n

b ` ÿ

d|b,d‰b

φ

´n d

¯

´1.

We now establish some inequalities involving degrees of vertices ofGpZ_nq.

4.2 Minimum degree 67

Proposition 4.2.6. Suppose n“p^α₁¹p^α₂². . . p^α_r^r where rě2, p₁ ăp₂ ă ¨ ¨ ¨ ăp_r are primes and α_i P N for 1ďiďr. Then the following inequalities hold in GpZ_nq.

(i) deg

´ p^α₁¹

¯

ědegpp^α_r^rq.

(ii) deg

´ p^γ_i

¯ ědeg

´ p^β_i

¯

for any 1ďiďr and 1ďγ ăβ ďα_i. (iii) deg

´ p^β_i

¯ ědeg

´ p^β_j

¯

for any 1ďiăj ďr and 1ďβ ďmintαi, αju.

(iv) deg

´

p^β₁¹p^β₂². . . p^βr^r

¯

ě deg

´

p^β₂². . . p^βr^r

¯ for

r

ÿ

i“1

β_i ă

r

ÿ

i“1

α_i, where 1 ď β_i ď α_i for any 1ďiďr.

Proof. (i) Let m “ n p^α₁¹p^α_r^r.

deg

´ p^α₁¹

¯

´degpp^α_r^rq “ n p^α₁¹ `

α1´1

ÿ

k“0

φ ˆn

p^k₁

˙

´

# n p^α_r^r `

αr´1

ÿ

k“0

φ ˆn

p^k_r

˙+

“φpmq

# φpp^α_r^rq

α1´1

ÿ

k“0

φ` p^α₁^1´k˘

´φpp^α₁¹q

αr´1

ÿ

k“0

φ` p^α_r^r´k˘

+

`mpp^α_r^r ´p^α₁¹q

“φpmq pp^α_r^r ´p^α_r^r´1qpp^α₁¹´1q ´ pp^α₁¹ ´p^α₁^1´1qpp^α_r^r ´1q(

`mtp^α₁¹pp^α_r^r ´1q ´p^α_r^rpp^α₁¹ ´1qu

“ pp^α_r^r ´1q“

p^α₁^1´1φpmq `p^α₁¹tm´φpmqu‰

´ pp^α₁¹ ´1q“

p^α_r^r´1φpmq `p^α_r^rtm´φpmqu‰ ě pp^α_r^r ´1q“

p^α₁^1´1φpmq `p^α₁¹tm´φpmqu‰

´p^α₁¹“

p^α_r^r´1φpmq `p^α_r^rtm´φpmqu‰

“ pp^α_r^rp^α₁^1´1´p^α₁¹p^α_r^r´1qφpmq ´“

p^α₁^1´1φpmq `p^α₁¹tm´φpmqu‰

“p^α₁^1´1“

p^α_r^r´1pp_r´p₁qφpmq ´ rφpmq `p<sub>1</sub>tm´φpmqus‰ ěp<sup>α</sup><sub>1</sub><sup>1´1</sup>rpp<sub>r</sub>´p<sub>1</sub>qφpmq ´ rφpmq `p₁tm´φpmquss

“p^α₁^1´1tpp_r´1qφpmq ´p₁mu. (4.1)

68 Minimum Degree and Connectivity

Now, if r“2, thenm“ n

p^α₁¹p^α₂² “1. Hence from (4.1), we get deg

´ p^α₁¹

¯

´degpp^α_r^rq ěp^α₁^1´1ppr´1´p1q ě0.

Ifr ą2, then m“ n p^α₁¹p^α_r^r “

r´1

ź

i“2

p^α_iⁱ. Again, from (4.1), we get

deg

´ p^α₁¹

¯

´degpp^α_r^rq ě

r´1

ź

i“1

p^α_i^i´1

# _r ź

i“2

pp_i´1q ´

r´1

ź

i“1

p_i +

ě0, since p_i`1 ´1ěp_i for all 1ďiďr´1.

(ii) deg

´ p^γ_i

¯

´deg

´ p^β_i

¯

“ n p^γ_i `

γ´1

ÿ

k“0

φ ˆn

p^k_i

˙

´ n p^β_i ´

β´1

ÿ

k“0

φ ˆn

p^k_i

˙

“ n p^γ_i ´ n

p^β_i ´

β´1

ÿ

k“γ

φ ˆn

p^k_i

˙

“ n p^α_iⁱ

´

p^α_i^i´γ´p^α_i^i´β

¯

´φ ˆ n

p^α_iⁱ

˙β´1

ÿ

k“γ

φ` p^α_i^i´k˘

“ n p^α_iⁱ

´

p^α_i^i´γ´p^α_i^i´β

¯

´φ ˆ n

p^α_iⁱ

˙´

p^α_i^i´γ´p^α_i^i´β

¯

“

´

p^α_i^i´γ´p^α_i^i´β

¯"

n p^α_iⁱ ´φ

ˆ n p^α_iⁱ

˙*

ě0,since γ ăβ.

(iii) deg

´ p^β_i

¯

´deg

´ p^β_j

¯

“ n p^β_i ´ n

p^β_j `

β´1

ř

k“0

# φ

ˆn p^k_i

˙

´φ

˜ n p^k_j

¸+

.

Sincep_i ăp_j, we have n p^β_i ą n

p^β_j . Furtherα_i, α_j ěβ, so that for all 0ďk ďβ´1, we haveφ

ˆn p^k_i

˙

´φ

˜ n p^k_j

¸

“

˜ n p^k_i ´ n

p^k_j

¸ r

ś

l“1

ˆ 1´ 1

p_l

˙

ě0. Hence the proof follows.

4.2 Minimum degree 69

(iv) deg

´

p^β₁¹p^β₂². . . p^βr^r

¯

´deg

´

p^β₂². . . p^βr^r

¯

“ n

p^β₁¹p^β₂². . . p^βr^r

`

ÿ

0ď

r

ř

i“1

γiă

r

ř

i“1

βi, 0ďγiďβi@1ďiďr

φ

ˆ n

p^γ₁¹p^γ₂². . . p^γr^r

˙

´

$

’’

’’

’&

’’

’’

’% n p^β₂². . . p^βr^r

`

ÿ

0ď řr

i“2

γiă řr

i“2

βi, 0ďγiďβi@2ďiďr

φ

ˆ n p^γ₂². . . p^γr^r

˙ , // // /. // // /-

“ n

p^β₁¹p^β₂². . . p^βr^r

`

ÿ

0ďγiďβi@1ďiďr

φ

ˆ n

p^γ₁¹p^γ₂². . . p^γr^r

˙

´φ

ˆ n

p^β₁¹p^β₂². . . p^βr^r

˙

´

# n p^β₂². . . p^βr^r

`

ÿ

0ďγiďβi@2ďiďr

φ

ˆ n p^γ₂². . . p^γr^r

˙

´φ

ˆ n p^β₂². . . p^βr^r

˙+

“

ÿ

1ďγ1ďβ1, 0ďγiďβi@2ďiďr

φ

ˆ n

p^γ₁¹p^γ₂². . . p^γr^r

˙

`φpmq

!

φpp^α₁¹q ´φpp^α₁^1´β1q )

`m

´

p^α₁^1´β1 ´p^α₁¹

¯ ´

by setting m “p^α₂^2´β2. . . p^α_r^r´βr

¯

ě

#

ÿ

0ďγiďβi@2ďiďr

φ`

p^α₂^2´γ2. . . p^α_r^r´γr˘ +

ÿ

1ďγ1ďβ1

φ`

p^α₁^1´γ1˘

`m

´

p^α₁^1´β1 ´p^α₁¹

¯

ěφpp^α₂². . . p^α_r^rq ÿ

1ďγ1ďβ1

φ`

p^α₁^1´γ1˘

`m

´

p^α₁^1´β1 ´p^α₁¹

¯

“

r

ź

i“2

p^α_i^i´1pp_i´1q( ÿ

1ďγ1ďβ1

φ`

p^α₁^1´γ1˘

´m

´

p^α₁¹ ´p^α₁^1´β1

¯

ěm

# _r ź

i“2

ppi´1q ÿ

1ďγ1ďβ1

φ`

p^α₁^1´γ1˘

´

´

p^α₁¹ ´p^α₁^1´β1

¯ +

.

Thus it is enough to show that ρ:“

r

ź

i“2

pp_i ´1q ÿ

1ďγ1ďβ1

φ`

p^α₁^1´γ1˘

´

´

p^α₁¹ ´p^α₁^1´β1

¯ ě0.

We have β1 ďα1. First take α1 “β1. Then ř

1ďγ1ďβ1

φ`

p^α₁^1´γ1˘

“p^α₁^1´1, so that

70 Minimum Degree and Connectivity

ρ“

r

ź

i“2

pp_i´1qp^α₁^1´1´ pp^α₁¹ ´1q ě pp₂´1qp^α₁^1´1´p^α₁¹ `1 ěp<sup>α</sup><sub>1</sub><sup>1</sup> ´p<sup>α</sup><sub>1</sub><sup>1</sup> `1ą0.

Now we takeα₁ ąβ₁. In this case, ř

1ďγ1ďβ1

φ`

p^α₁^1´γ1˘

“p^α₁^1´1´p^α₁^1´β1´1, so that

ρ“ pp₂´1q. . .pp_r´1q

´

p^α₁^1´1´p^α₁^1´β1´1

¯

´

´

p^α₁¹ ´p^α₁^1´β1

¯

“

´

p^α₁^1´1 ´p^α₁^1´β1´1

¯

tpp₂´1q. . .pp_r´1q ´p₁u ě0.

Theorem 4.2.7. Let nPN and p₁ ăp₂ ăp₃ ăp₄ be prime numbers.

(i) If n“p^α₁¹p^α₂², α₁, α₂ PN, then p^α₂² has the minimum degree among all vertices in GpZ_nq, and δpGpZ_nqq “ pp^α₂² ´1qφpp^α₁¹q `p^α₁¹ ´1.

(ii) If n “ p₁p₂p₃, then p₃ has the minimum degree among all vertices in GpZ_nq, and δpGpZ_nqq “φpnq `p₁p₂´1.

(iii) Let n “ p₁p₂p₃p₄. If n is odd or p₄ ě p₃ ` 2pp3´1q

p₂´1 , then p₄ has the mini- mum degree among all vertices in GpZ_nq, and δpGpZ_nqq “ φpnq `p₁p₂p₃ ´1.

Otherwise, p₃p₄ has the minimum degree among all vertices in GpZ_nq, and δpGpZ_nqq “ pp₂´1qpp₃p₄`1q `1.

Proof. In view of Lemma 4.2.3, to determine δpGpZ_nqq, it is sufficient to compare the degrees of vertices of the form c, wherecą1 is a proper divisor of n.

(i) Consider β1, β2 P N satisfying 1 ď βi ď αi for i “1,2. Using Proposition 4.2.6, we have

4.2 Minimum degree 71

(a) deg

´ p^β₁¹

¯ ědeg

´ p^α₁¹

¯ ědeg

´ p^α₂²

¯ .

(b) deg

´ p^β₂²

¯ ědeg

´ p^α₂²

¯ .

(c) deg

´ p^β₁¹p^β₂²

¯ ědeg

´ p^β₂²

¯ ědeg

´ p^α₂²

¯ .

Hence p^α₂² has the minimum degree among all vertices in GpZnqand by Lemma 4.2.5, δpGpZ_nqq “ pp^α₂² ´1qφpp^α₁¹q `p^α₁¹ ´1.

(ii) If i, j, k is a permutation of 1,2,3 with iăj, we have degpp_ip_jq ´degpp_jq “p_k`φpp<sub>i</sub>p<sub>k</sub>q `φpp_jp_kq ´p_ip_k

“ pp_i´1qpp_k´1q ` pp_j ´1qpp_k´1q ´p_kpp_i´1q

“ pp_j ´1qpp_k´1q ´ pp_i´1q ě 0, since p_i ăp_j.

Further, by Proposition 4.2.6(iii), degpp₁q ě degpp₂q ě degpp₃q. Hence p₃ has the minimum degree among all vertices in GpZ_nq. Consequently, by Lemma 4.2.5, δpGpZ_nqq “ degpp₃q “ φpnq `p₁p₂´1.

(iii) Let i, j, k, l be a permutation of 1,2,3,4.

Foriăj ăk, we have

degpp_ip_jp_kq ´degpp_jp_kq

“p_l` ÿ

dpipjpk

d‰pipjpk

φ

´n d

¯

´

$

’’

&

’’

%

p_ip_l` ÿ

dpjpk

d‰pjpk

φ

´n d

¯ , // . // -

“p_l`φ ˆ n

p_ip_j

˙

`φ ˆ n

p_ip_k

˙

`φ ˆ n

p_jp_k

˙

`φ ˆn

p_i

˙

´p_ip_l

72 Minimum Degree and Connectivity

“ pp_l´1q tpp_k´1q ` pp<sub>j</sub> ´1q ` pp_i´1q ` pp_j´1qpp_k´1qu ´p_lpp_i´1q

“ pp_l´1q tpp_k´1q ` pp<sub>j</sub> ´1q ` pp_j ´1qpp_k´1qu ´ pp_i´1q ě pp_l´1qpp_j ´1q ´ pp_i´1q ě0, since p_i ăp_j.

Now take iăj with no condition on k and l.

degpp_ip_jq ´degpp_jq “p_kp_l`φpp<sub>i</sub>p<sub>k</sub>p<sub>l</sub>q `φpp_jp_kp_lq ´p_ip_kp_l

“ pp_i´1qpp_k´1qpp_l´1q ` pp_j ´1qpp_k´1qpp_l´1q ´p_lp_kpp_i´1q

“ pp_j´1qpp_k´1qpp_l´1q ´ pp_i´1qpp_k`p_l´1q. (4.2)

Since k and l can be interchanged in (4.2), without loss of generality, let pk ăpl. Ifn is odd, then p_k ą2, so that

degppipjq ´degppjq

ě ppi´1qtppk´1qppl´1q ´ ppk`pl´1qu psince pi ăpjq

“ ppi´1qtppk´2qppl´1q ´pku

ě ppi´1qtppl´1q ´pku ě 0, since pk ą2 and pkăpl.

Now letnbe even, that is,p₁ “2. Ifp_ką2, then from (4.2), degpp_ip_jq ědegpp_jq as shown above. So take p_k “2. From (4.2), we have

degpp_ip_jq ´degpp_jq “ pp_j ´1qpp_l´1q ´ pp_i´1qpp_l`1q. (4.3)

In (4.3), letpi ‰p3. Since iăj,pi cannot be p4. Moreover, pk “p1 “2, so we have p_i “p₂. As a result, p_l ąp₂. Then from (4.3),

4.2 Minimum degree 73

degpp_ip_jq ´degpp_jq “ pp_l´1qtpp_j ´1q ´ pp₂´1qu ´2pp₂´1q ě2pp_l´1q ´2pp₂´1q (since p_j´p₂ ě2q

“2pp_l´p₂q ą0, since p_ląp₂.

Now take p_i “ p₃ in (4.3). Then p_j “ p₄. We already have p_k “ 2, and since p_k ăp_l, we have p_l “p₂. Then from (4.3),

degpp_ip_jq ´degpp_jq “degpp₃p₄q ´degpp₄q “ pp₄´1qpp₂´1q ´ pp₃´1qpp₂`1q, and hence

degpp₃p₄q ě degpp₄q ôp₄ ěp₃`2pp3´1q

p₂´1 . (4.4)

Case 1: n is odd or p₄ ěp₃` 2pp₃´1q p2´1

As shown above, for all 1ďiăj ăkď4,

degpp_ip_jp_kq ědegpp_jp_kq ě degpp_kq. (4.5) Further, it follows from Proposition 4.2.6(iii) that

degpp₁q ědegpp₂q ě degpp₃q ě degpp₄q. (4.6) So we conclude that p₄ has the minimum degree among all vertices in GpZ_nq. Con- sequently, by Lemma 4.2.5, we get δpGpZ_nqq “degpp₄q “ φpnq `p₁p₂p₃´1.

Case 2: n is even and p₄ ăp₃` 2pp₃´1q p₂´1

Then from (4.4), degpp3p4q ădegpp4q, whereas all other inequalities in (4.5) and (4.6) hold. Thus p₃p₄ has the minimum degree among all vertices in GpZ_nq. Thus

74 Minimum Degree and Connectivity

by Lemma 4.2.5, we get

δpGpZ_nqq “2p₂` pp<sub>2</sub>´1qtpp<sub>3</sub>´1qpp<sub>4</sub>´1q ` pp₃´1q ` pp₄´1qu ´1

“ pp₂´1qpp₃p₄`1q `1.

Corollary 4.2.8. Let n “ p^α₁¹p^α₂². . . p^α_r^r, r ě 2, p₁ ă p₂ ă ¨ ¨ ¨ ă p_r be prime numbers and αi P N for 1ďiďr. Let

η₁pnq “ n

p^α_r^r ` pp^α_r^r ´1qφ ˆ n

p^α_r^r

˙

´1, (4.7)

and

η₂pnq “ n

p_r´1p_r `φpnq `φ ˆn

p_r

˙

`φ ˆ n

p_r´1

˙

´1. (4.8)

Then η₁pnq and η₂pnq are sharp upper bounds of δpGpZ_nqq.

Proof. By Lemma 4.2.5,

degpp^α_r^rq “ η₁pnq, (4.9)

and

degpp_r´1p_rq “η₂pnq, (4.10) so that η₁pnq and η₂pnq are upper bounds of δpGpZ_nqq. Moreover, as shown in Theorem 4.2.7, δpGpZ_nqqattainsη₁pnqfor some values ofn andη₂pnqfor some other values of n, and hence the bounds are sharp.

In view of Lemma 4.2.1 and Lemma 4.1.3, we have the following corollary of Theorem 4.2.7.

Corollary 4.2.9. Let nP N and p₁ ăp₂ ăp₃ ăp₄ be prime numbers.

(i) If n “p^α₁¹p^α₂², α₁, α₂ PN, then for any a P

” p^α₂²

ı , E

„ a,

A p^α₂²

E Y

α2´1

Ť

i“0

” pⁱ₂

ı

´a



is a minimum disconnecting set of GpZ_nq.

4.2 Minimum degree 75

(ii) If n “ p₁p₂p₃, then for any a P rp₃s, E“

a,xp₃y Y“ 1‰

´a‰

is a minimum dis- connecting set of GpZ_nq.

(iii) Let n “ p₁p₂p₃p₄. If n is odd or p₄ ě p₃ ` 2pp3´1q

p₂´1 , then for any a P rp₄s, E“

a,xp₄y Y“ 1‰

´a‰

is a minimum disconnecting set of GpZ_nq. Otherwise, for anyb P rp₃p₄s,E“

b,xp₃p₄y Y rp₃s Y rp₄s Y“ 1‰

´b‰

is a minimum disconnecting set of GpZ_nq.