Vertex connectivity of GpZ n q

34 Vertex Connectivity

We begin this section with the following lemma that provides some basic and useful properties of Z_n.

Lemma 2.3.1. For n PN, the following statements hold for «-classes of Z_n. (i) For each aPZ^˚_n, there exists a positive divisor d of n such that a«d.

(ii) For aPZ^˚_n, |ras| “φ

ˆ n gcdpn, aq

˙ .

(iii) If a, bPN is such that an, bn and a‰b, then affb.

Proof. (i) Consider d“gcdpa, nq. Then xay “@ dD

so that a«d.

(ii) Since opaq “ n

gcdpn, aq, the proof follows from Corollary 1.3.23.

(iii) Suppose xay “ xby. Then opaq “opbq. That is, n

gcdpn, aq “ n

gcdpn, bq. As an and bn, we in turn get a“b; which is a contradiction. Hence a ffb.

It is known thatGpZ_nqis a complete graph whennis a prime power (cf. Theorem 1.5.7(ii)). We now focus on connectedness of GpZ_nqwhen n is not a prime power.

Lemma 2.3.2. If n ą1 is not a prime number, then the following statements hold.

(i) If n is not a prime power, then every separating set of GpZnq contains SpZnq.

(ii) κpGpZ_nqq “ φpnq `1`κpG¹pZ_nqq.

(iii) If p₁ ăp₂ ă ¨ ¨ ¨ ăp_r are the prime factors of n, then Z¹_n “

r

ď

i“1

xp_iy^˚.

Proof. When nis not a prime power, GpZ_nqis not complete (cf. Theorem 1.5.7). So (i) follows from Remark 1.5.14. Since n is not a prime number,G¹pZnq is a non-null graph. So (ii) follows from Remark 1.5.14 and the fact that |SpZ_nq| “ φpnq `1.

Moreover, if aP Z¹_n, then a is divided by at least one prime factor of n. Hence (iii) follows.

2.3 Vertex connectivity of GpZ_nq 35

Ifn is a product of two distinct primes, then κpGpZ_nqq “φpnq `1 (cf. Theorem 1.5.19(ii)). In the next result, we show that the converse holds as well.

Proposition 2.3.3. For nPN, the following statements are equivalent.

(i) n is a product of two distinct primes.

(ii) SpZ_nq is a separating set of GpZ_nq. (iii) κpGpZ_nqq “φpnq `1.

Proof. It follows from Theorem 1.5.19(ii) that (i) implies (iii). Whereas, if (iii) holds, since |SpZ_nq| “ φpnq `1, (ii) follows from Lemma 2.3.2(i).

We now prove that (ii) implies (i). LetSpZ_nqbe a separating set ofGpZ_nq. Then GpZ_nq is not a complete graph and hence by Theorem 1.5.7(ii), n has at least two distinct prime factors. Since G¹pZ_nq is disconnected, there exist a, bP Z¹_n such that there is no path from atob inG¹pZ_nq. In view of Lemma 2.3.2(iii), every element of Z¹_n is in xpy^˚ for some prime factor p of n. Let a, bP xpy for some prime factor p of n. Then a, p, bis ana, b-path inG¹pZ_nq, which is a contradiction. HenceaP xpy and b P xqy for some distinct prime factors p and q of n. If possible, letpq ăn, so that pq P Z¹_n. Consequently, a and b are connected by the path a, p, pq, q, b in G¹pZ_nq, which is again a contradiction. Hence n“pq.

Theorem 2.3.4. Suppose n is not a product of two primes and has prime factors p₁ ă p₂ ă ¨ ¨ ¨ ă p_r with r ě 2. Then, for any 1 ď k ď r,

r

ď

i“1i‰k

xp_ip_ky^˚ is a minimal separating set of G¹pZ_nq.

Proof. Let Γ “G¹pZnq. Then by Proposition 2.3.3, Γ is connected and by Lemma 2.3.2(iii), VpΓq “

r

ď

i“1

xp_iy^˚. Let T “

r

ď

i“1i‰k

xp_ip_ky^˚. For i “1,2, . . . , r, letT_i “ xp_iy^˚´T and U “

r

ď

i‰ki“1

T_i. Then

36 Vertex Connectivity

VpΓ´Tq “

r

ď

i“1

T_i “T_kYU. We prove that Γ´T is disconnected by showing that no element of T_k is adjacent to any element of U.

If possible, let there existxPTkandyP U such that they are adjacent. SoyPTl

for some 1 ď l ď r, l ‰ k. Then there exist non-zero integers a and b such that x“apk andy“bpl. Sincexis adjacent to y, one of them is a multiple of the other.

Let ap_k “ cbp_l for some non-zero integer c. Then ap_k “ cbp_l `c₁n for some non- zero integer c1. This implies that plapk. Since plfflpk, we have pla. Consequently, ap_k P xp_kp_ly Ď T. This is a contradiction, as T_k XT “ H. Similarly, if bp_l is a multiple of apk, then also we get a contradiction. Hence Γ´T is disconnected.

Consequently, T is a separating set of Γ.

We now show the minimality ofT. First of all,GpTkqis connected because all of its vertices are adjacent top_k. Moreover, note thatGpUqis connected. For instance, let u, v P U. Then u P Ti, v P Tj for some 1 ďi, j ď r. If i “ j, then u and v are connected by the path u, p_i, v, and ifi‰j, thenu andv are connected by the path u, pi, pipj, pj, v.

Now let z P T. So z “ dp_kp_l for some non-zero integer d and 1 ď l ď r, l ‰ k. Since both pk P Tk and pl P U are adjacent to z, Γ´ pT ´zq is connected.

Consequently, T is a minimal separating set of G¹pZ_nq.

Lemma 2.3.5. Suppose n is not a product of two primes and n “ p^α₁¹p^α₂². . . p^α_r^r, where r ě2, p1 ăp2 ă ¨ ¨ ¨ ăpr are primes and αi P N for all 1ďi ďr. Then for any 1ďk ďr,

ˇ ˇ ˇ ˇ ˇ ˇ ˇ

r

ď

i“1i‰k

xp_ip_ky ˇ ˇ ˇ ˇ ˇ ˇ ˇ

“ n

p_k ´p^α_k^k´1φ ˆ n

p^α_k^k

˙

and for any 1ďj ăk ďr, ˇ ˇ ˇ ˇ ˇ ˇ ˇ

r

ď

i“1i‰k

xp_ip_ky ˇ ˇ ˇ ˇ ˇ ˇ ˇ

ď ˇ ˇ ˇ ˇ ˇ ˇ ˇ

r

ď

i“1i‰j

xp_ip_jy ˇ ˇ ˇ ˇ ˇ ˇ ˇ

. (2.3)

2.3 Vertex connectivity of GpZ_nq 37

Proof. Observe that

r

ď

i“1 i‰k

xp_ip_ky consists of those elements of xp_ky which are divisible by some p_i, 1ďiďr, i‰k. Therefore, we get

r

ď

i“1 i‰k

xp_ip_ky by deleting those elements fromxp_kywhich are relatively prime top_i, 1ďiďr, i ‰k. Hence

r

ď

i“1i‰k

xp_ip_ky “A´B, where A“

"

ap_k :aPN,0ďa ă n p_k

* , and B “

"

ap_k :a PN,0ďaă n pk

,pa, p_iq “ 1@1ďiďr, i ‰k

* . Take n1 “ śr

j“1,j‰kpj and n2 “ n

p_kn₁ “ n

p₁. . . p_r. For 0 ď m ď n2 ´1, let P_m “ tap_k :aPN, mn₁ ďaă pm`1qn₁,pa, n₁q “ 1u. Trivially P_l XP_m “ H for l ‰m and

B “

n2´1

ď

m“0

P_m. (2.4)

Observe thatap_kP P_m if and only if pa´mn₁qp_kP P₀. Thus |P_m| “ |P₀| for all 0ďm ďn2´1. Further, |P0| “ |tapk :aPN,0ďa ăn1,pa, n1q “ 1u| “φpn1q. So for all 0ďmďn₂´1,

|Pm| “φpn1q. (2.5)

From (2.4) and (2.5), we have

|B| “

n2´1

ÿ

m“0

|P_m| “n₂φpn₁q “ n p_k

r

ź

i“1 i‰k

ˆ 1´ 1

p_i

˙

“p^α_k^k´1φ ˆ n

p^α_k^k

˙ .

As |A| “ n

p_k and B ĎA, we finally have ˇ

ˇ ˇ ˇ ˇ ˇ ˇ

r

ď

i‰ki“1

xp_ip_ky ˇ ˇ ˇ ˇ ˇ ˇ ˇ

“ |A| ´ |B| “ n

p_k ´p^α_k^k´1φ ˆ n

p^α_k^k

˙ .

38 Vertex Connectivity

Now we prove (2.3).

ˇ ˇ ˇ ˇ ˇ ˇ ˇ

r

ď

i“1i‰j

xp_ip_jy ˇ ˇ ˇ ˇ ˇ ˇ ˇ

´ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

r

ď

i‰ki“1

xp_ip_ky ˇ ˇ ˇ ˇ ˇ ˇ ˇ

“ n

p_j ´p^α_j^j´1φ

˜ n p^α_j^j

¸

´

"

n

p_k ´p^α_k^k´1φ ˆ n

p^α_k^k

˙*

“ n p_j ´ n

p_j

r

ź

i“1i‰j

ˆ 1´ 1

p_i

˙

´

$

’&

’% n p_k ´ n

p_k

r

ź

i“1i‰k

ˆ 1´ 1

p_i

˙ , /. /-

“ n p_jp_k

»

—

–p_k´p_k

r

ź

i“1 i‰j

ˆ 1´ 1

p_i

˙

´

$

’&

’%

p_j ´p_j

r

ź

i“1 i‰k

ˆ 1´ 1

p_i

˙ , /. /-

fi ffi fl

“ n p_jp_k

$

’&

’%

p_k´p_j ´ tp_k´1´ pp_j ´1qu

r

ź

i‰j,ki“1

ˆ 1´ 1

p_i

˙ , /. /-

“ npp_k´p_jq p_jp_k

$

’&

’% 1´

r

ź

i‰j,ki“1

ˆ 1´ 1

p_i

˙ , /. /-

ě0.

In the following theorem, we shall give an upper bound for the vertex connectivity of power graphs of Z_n. This generalizes Theorem 1.5.20 due to Chattopadhyay and Panigrahi [2014] to all nPN.

Theorem 2.3.6. If n “ p^α₁¹p^α₂². . . p^α_r^r, where r ě2, p₁ ăp₂ ă ¨ ¨ ¨ ă p_r are primes and α_j PN for 1ďj ďr, then

κpGpZ_nqq ďφpnq ` n

p_r ´p^α_r^r´1φ ˆ n

p^α_r^r

˙

. (2.6)

Proof. Ifn is a product of two primes, the inequality follows from Proposition 2.3.3.

Now suppose nis not a product of two primes. By Theorem 2.3.4 and Lemma 2.3.5, we have

κpG¹pZ_nqq ď ˇ ˇ ˇ ˇ ˇ

r´1

ď

i“1

xp_ip_ry^˚ ˇ ˇ ˇ ˇ ˇ

“ n

p_r ´p^α_r^r´1φ ˆ n

p^α_r^r

˙

´1.

2.3 Vertex connectivity of GpZ_nq 39

Hence the proof follows from Lemma 2.3.2(ii).

Remark 2.3.7. We prove in Theorem 2.3.20 and Theorem 2.3.22 that the equality holds in (2.6) if n has exactly two prime factors or n is a product of three distinct primes. This will show that the upper bound given in Theorem 2.3.6 is sharp.

We now concentrate on minimal separating sets of GpZ_nq that arise from neigh- borhoods of «-classes. Subsequently, we obtain an alternate upper bound for κpGpZ_nqq, and ascertain the conditions on n for which this bound is an improve- ment to that of Theorem 2.3.6.

The following remark is a consequence of Lemma 2.2.7.

Remark 2.3.8. If a P SpZ_nq, then Npaq and Nprasq are not separating sets of GpZ_nq.

The next remark is immediate from Lemma 2.1.6.

Remark 2.3.9. rSpZ_nqs “ tr0s,r1su. Notice that for any a P Z¹_n, we haveSpZ_nq Ď Npaq and rSpZ_nqs ĎNrprasq.

Notation 2.3.10. We denote Gr¹pZ_nq “ GrpZ_nq ´ rSpZ_nqs, N¹paq “ Npaq ´SpZ_nq, N¹prasq “Nprasq ´SpZ_nq and Nr¹prasq “Nrprasq ´ rSpZ_nqs.

Remark 2.3.11. For anyn PN,Gr¹pZ_nq “ GrpZ_n´SpZ_nqq.

Lemma 2.3.12. Suppose n P N is neither a prime power nor a product of two distinct primes. Then for any a PZ¹_n,

(i) N¹prasq is a separating set of G¹pZ_nq.

(ii) Nr¹prasq is a separating set of Gr¹pZnq.

(iii) N¹prasqis a minimal separating set ofG¹pZ_nqif and only ifNr¹prasqis a minimal separating set of Gr¹pZnq.

40 Vertex Connectivity

Proof. By Theorem 1.5.7, GpZ_nq is not complete and by Proposition 2.3.3, G¹pZ_nq is connected. Thus by Lemma 2.2.7, Nprasq is a separating set ofGpZ_nqand Nrprasq is a separating set of GpZr _nq. Consequently, (i) and (ii) follow from Remark 1.5.14.

Finally, (iii) follows from (i) and Theorem 2.2.4.

Lemma 2.1.11 and Lemma 2.3.1 give the following observation, which will be useful in proving Theorem 2.3.14.

Lemma 2.3.13. For nPN, a PZ¹_n andb “gcdpa, nq, the following holds in GpZ_nq:

N¹prasq “ ď

cb 1ăcăb

rscs Y ď

bd,dn bădăn

rds.s (2.7)

The following theorem further narrows down our study of minimal separating sets of GpZ_nq.

Theorem 2.3.14. SupposenPNis not a product of two primes andn “p^α₁¹p^α₂². . . p^α_r^r, where r ě2, p1 ăp2 ă ¨ ¨ ¨ ăpr are primes and αi P N for all 1ďi ďr. Then for any 1ďk ďr, the following statements hold.

(i) N¹

´”

p^α_k^k ı¯

is a minimal separating set of G¹pZ_nq.

(ii) If α_k ą 1 and 1 ď β_k ă α_k, N¹ ˆ„

p^β_k^k

˙

is not a minimal separating set of G¹pZnq.

Proof. (i) Let T “ Nr¹

´”

p^α_k^k ı¯

and Γ “ Gr¹pZnq ´T. By Lemma 2.3.12(ii), Γ is disconnected. Moreover, by Lemma 2.1.11,VpΓq “

!”

p^α_k^k ı)Ť

tras:an,1ăaăn, p^α_k^kffla, afflp^α_k^ku. Let T1 “

!”

p^α_k^k ı)

and T2 “ tras:an,1ăaăn, p^α_k^kffla, afflp^α_k^ku.

Then the subgraph of Γ induced by T₁ is complete and hence connected. Further- more, rpis P T2 for all 1ďi ďr, i ‰k and every other rbs P T2 is adjacent to some rp_js for 1 ď j ď r, j ‰ k in Γ. Additionally, rp_is,rp_js P T₂, i ‰ j, then both are adjacent torpipjs PT2in Γ. Thus the subgraph of Γ induced byT2 is also connected.

So Γ consists of exactly two components - the subgraphs induced by T₁ and T₂.

2.3 Vertex connectivity of GpZ_nq 41

Therefore, to show thatT is a minimal separating set of Gr¹pZ_nq, it is enough to show that every element of T is adjacent to some element of T₁ and some element of T₂. Let C PT. Then

” p^α_k^k

ı

is adjacent toC. We next show thatC is adjacent to some element of T₂.

By Lemma 2.3.1(i), C “ rcs for some 1 ă c ă n, cn. Since C is adjacent to

” p^α_k^k

ı

, it follows again from Lemma 2.1.11 that either p^α_k^kˇ

ˇc or cˇ

ˇp^α_k^k. As both c and p^α_k^k are factors of n, either p^α_k^kˇ

ˇc or cˇ

ˇp^α_k^k. First let p^α_k^kˇ

ˇc, so that c “ ap^α_k^k for some integer a. If gcd

ˆ a, n

p^α_k^k

˙

“ 1, then C “

” p^α_k^k

ı

, which is not possible.

So, as n p^α_k^k “

r

ś

i“1,i‰k

p^α_iⁱ, there exist 1 ďl ď r, l‰ k such that p_la. Then p_la and hence p_lc. So C is adjacent to rp_ls and rp_ls P T₂. Now let cˇ

ˇp^α_k^k. Then c “ p^β_k for some 1 ď β ă α_k. So C is adjacent to

” p^β_kp_m

ı

for all 1 ď m ď r, m ‰ k and

” p^β_kp_m

ı

P T₂. Thus T is a minimal separating set of Gr¹pZ_nq and hence the proof follows from Lemma 2.3.12(iii).

(ii) Observe that

” p^α_k^k

ı PNr¹

´”

p^β_k^k ı¯

. So that by Lemma 2.3.13, we have

Nr¹

´”

p^α_k^k ı¯

ĎNr¹

´”

p^β_k^k ı¯

Y

!”

p^β_k^k ı)

. (2.8)

By Lemma 2.1.11,

” p^β_k^k

ı

is adjacent to

” p^α_k^k

ı

, and by (2.8), none of

” p^α_k^k

ı and

” p^β_k^k

ı

is adjacent to any other vertex of Gr¹pZ_nq ´

´ Nr¹

´”

p^β_k^k ı¯

´

” p^α_k^k

ı¯

. Hence Nr¹

´”

p^β_k^k ı¯

´

” p^α_k^k

ı

is a separating set of Gr¹pZ_nq and as a result, Nr¹

´”

p^β_k^k ı¯

is not a minimal separating set of Gr¹pZ_nq. From this and Lemma 2.3.12(iii), the proof follows.

The next result shows that minimal separating sets ofGpZnqobtained in Theorem 2.3.4 and Theorem 2.3.14 are same when the largest prime dividingnhas power one.

Proposition 2.3.15. Suppose n P N is not a product of two primes and n “ p^α₁¹. . . p^α_r´1^r´1p_r, where rě2, p₁ ăp₂ ă ¨ ¨ ¨ ăp_r are primes and α_i P Nfor 1ďiďr.

Then N¹prp_rsq “Ťr´1

i“1xp_ip_ry^˚.

42 Vertex Connectivity

Proof. N¹prp_rsq “ xp_ry^˚´ rp_rs

“ xp_ry^˚´ ap_r : 1ďaăp^α₁¹. . . p^α_r´1^r´1,gcdpa, p^α₁¹. . . p^α_r´1^r´1q “ 1(

“

r´1

ď

i“1

xp_ip_ry^˚

We now provide an upper bound forκpGpZ_nqq in the following theorem.

Theorem 2.3.16. Suppose n is not a product of two primes and n “p^α₁¹p^α₂². . . p^α_r^r, where rě2, p1 ăp2 ă ¨ ¨ ¨ ăpr are primes and αi PN for 1ďiďr, then

κpGpZ_nqq ďφpnq ` n

p^α_r^r ` pp^α_r^r´1´2qφ ˆ n

p^α_r^r

˙ .

Proof. From Theorem 2.3.14(i), κpG¹pZ_nqq ď |N¹prp^α_r^rsq|, and by Lemma 2.3.13, we have

|N¹prp^α_r^rsq| “ˇ

ˇxp^α_r^ry^˚ˇ

ˇ´ |rp^α_r^rs| `

αr

ÿ

j“1

ˇ ˇ ˇ

” p^jr

ıˇ ˇ

ˇ´ |rp^α_r^rs|

“ n

p^α_r^r ´1´φ ˆ n

p^α_r^r

˙

`

αr

ÿ

j“1

φ ˆn

p^jr

˙

´φ ˆ n

p^α_r^r

˙

“ n

p^α_r^r ´1`φ ˆ n

p^α_r^r

˙ αr

ÿ

j“1

φ` p^α_r^r´j˘

´2φ ˆ n

p^α_r^r

˙

“ n

p^α_r^r ` pp^α_r^r´1´2qφ ˆ n

p^α_r^r

˙

´1.

Hence by Lemma 2.3.2(ii), the proof follows.

Notation 2.3.17. We denote the upper bound of κpGpZ_nqq obtained in Theorem 2.3.6 and Theorem 2.3.16 by ξ₁pnqand ξ₂pnq, respectively.

In the following theorem, we compare the upper bounds ξ1pnq and ξ2pnq of κpGpZ_nqqin the following theorem.

2.3 Vertex connectivity of GpZ_nq 43

Theorem 2.3.18. Supposen is not a product of two primes and n “p^α₁¹p^α₂². . . p^α_r^r, where rě2, p₁ ăp₂ ă ¨ ¨ ¨ ăp_r are primes and α_i P N for 1ďiďr.

(i) ξ₂pnq “ξ₁pnq if and only if α_r “1, or r“2 and p₁ “2.

(ii) ξ₂pnq ăξ₁pnq if and only if α_r ě2 and

r´1

ś

i“1

ˆ 1´ 1

pi

˙ ă 1

2. (iii) ξ₂pnq ąξ₁pnq if and only if α_r ě2 and

r´1

ś

i“1

ˆ 1´ 1

p_i

˙ ą 1

2. Proof. Note that

ξ₂pnq ´ξ₁pnq “ n

p^α_r^r ` pp^α_r^r´1´2qφ ˆ n

p^α_r^r

˙

´

"

n

p_r ´p^α_r^r´1φ ˆ n

p^α_r^r

˙*

“`

1´p^α_r^r´1˘ n

p^α_r^r `2pp^α_r^r´1´1qφ ˆ n

p^α_r^r

˙

“ pp^α_r^r´1´1q

"

2φ ˆ n

p^α_r^r

˙

´ n p^α_r^r

*

“ pp^α_r^r´1´1q n p^α_r^r

# 2

r´1

ź

i“1

ˆ 1´ 1

p_i

˙

´1 +

. (2.9)

The right hand side of (2.9) equals 0 if and only ifα_r“1, or

2

r´1

ź

i“1

ˆpi´1 p_i

˙

“1. (2.10)

We show that (2.10) holds if and only if r “2 and p₁ “2.

If p₁ ą 2, then

r´1ś

i“1

p_i is odd. Since 2

r´1ś

i“1

pp_i´1q is always even, (2.10) does not hold. Thus p₁ “2. If r ą2, then

2

r´1

ź

i“1

ˆp_i´1 p_i

˙

“

r´1

ź

i“2

ˆp_i´1 p_i

˙

‰1

as the numerator is even and denominator is odd. So we must have r “ 2. Con- versely, if r “2 and p₁ “2, then (2.10) holds. This proofs (i).

Since p^α_r^r´1´1ą0 if and only ifαrě2, (ii) and (iii) follow from (2.9).

44 Vertex Connectivity

Remark 2.3.19. Thelexicographic order ăonNˆN, defined bypa₁, b₁qăpa₂, b₂q if

a₁ ăa₂, ora₁ “a₂ and b₁ ăb₂, is a well-founded relation.

We now give the actual values of κpGpZ_nqq when n has two prime factors (cf.

Theorem 2.3.20) and n is a product of three distinct primes (cf. Theorem 2.3.22).

Theorem 2.3.20. If n“p^αq^β, where p, q are distinct primes and α, β P N, then κpGpZ_nqq “ φpnq `p^α´1q^β´1. (2.11) In fact, for n ‰pq, xpqy^˚ is a minimum separating set of G¹pZ_nq.

Proof. We consider the lexicographic order ă onNˆN, and prove by applying the principle of well-founded induction that (2.11) holds for all pα, βq P NˆN. Note that, as n is not a prime power, GpZnq and hence G¹pZnqare not complete graphs.

Since κpGpZ_pqqq “ φppqq ` 1 (cf. Theorem 1.5.19), the statement holds for pα, βq “ p1,1q.

Now take pα, βq P NˆN such that p1,1q ă pα, βq. Suppose that (2.11) holds for all pa, bqă pα, βq. Then n ‰pq and hence by Proposition 2.3.3, Γ :“G¹pZ_nq is connected. Further, by Theorem 2.3.4, xpqy^˚ is a minimal separating set of Γ. We show that xpqy^˚ is a minimum separating set of Γ.

LetT be a minimal separating set of Γ. We show that|xpqy^˚| ď |T|. Ifxpqy^˚ ĎT, we are done. So let xpqy^˚ Ć T. Then there exists an element a P xpqy^˚ such that a R T. Let Γ₁ “ Gpxpy^˚q and Γ₂ “ Gpxqy^˚q. Then observe that Γ “ Γ₁ YΓ₂, and hence Γ´T “ pΓ₁´Tq Y pΓ₂´Tq. Further, aPVpΓ₁´Tq XVpΓ₂´Tq. Hence as Γ´T is disconnected, at least one of Γ₁´T or Γ₂´T is disconnected.

Case 1: Let Γ₁ ´T be disconnected. If α “ 1, then |xpy| “ q^β and hence Γ₁ is a

2.3 Vertex connectivity of GpZ_nq 45

complete graph. So Γ₁´T cannot be disconnected. So αě2. Then,

|T| ´ |xpqy^˚| ěκpΓ₁q ´ |xpqy^˚|

“κpGpxpy^˚qq ´ |xpqy^˚|

“κpGpxpyq ´0q ´ |xpqy^˚|

“κpGpxpyqq ´1´ p|xpqy| ´1q pby Remark 1.5.11q

“κpGpZ_pα´1q^βqq ´ |xpqy|

“φpp^α´1q^βq `p<sup>α´2</sup>q<sup>β´1</sup>´`

p^α´1q^β´1˘

(by induction hypothesis)

“p^α´2q^β´1pp´1qpq´1q `p^α´2q^β´1´p^α´1q^β´1

“p^α´2q^β´1tpp´1qpq´1q `1´pu

“p^α´2q^β´1pp´1qpq´2q ě0. (2.12)

Case 2: Let Γ2´T be disconnected. Proceeding as in Case 1, we have β ě2, and

|T| ´ |xpqy^˚| ěκpΓ₂q ´ |xpqy^˚|

“κpGpxqy^˚qq ´ |xpqy^˚|

“p^α´1q^β´2tpp´1qpq´1q `1´qu

ěp^α´1q^β´2tpq´1q `1´qu “0. (2.13) So for p1,1q ă pα, βq, xpqy<sup>˚</sup> is a minimum separating set of G<sup>1</sup>pZ<sub>n</sub>q and hence κpGpZ<sub>n</sub>qq “φpnq `p^α´1q^β´1. Therefore by the principle of well-founded induction, (2.11) holds for all pα, βq PNˆN.

The following is a simple consequence of Theorem 2.3.20.

Corollary 2.3.21. If n “ 2^αp^β, where p is an odd prime and α, β P N, then κpGpZ_nqq “ n

2.

Proof. κpGpZ_nqq “φpnq`2<sup>α´1</sup>p<sup>β´1</sup> “2<sup>α´1</sup>p<sup>β´1</sup>pp2´1qpp´1q`1q “2^α´1p^β “ n 2.

46 Vertex Connectivity

In the following result, we obtain the vertex connectivity of GpZ_nq when n is a product of three distinct primes.

Theorem 2.3.22. If n “ pqr, where p ă q ă r are primes, then rprs Y rqrs is a minimum separating set of G¹pZ_nq. Consequently,

κpGpZ_nqq “ φpnq `p`q´1.

Proof. It follows from Lemma 2.3.1 that the equivalence classes of Z¹_n with respect to «are rps, rqs,rrs, rpqs, rprs and rqrs.

rps

rprs rpqs

rqrs

rrs rqs

Figure 2.1: Gr¹pZpqrq

Using Remark 2.1.12,Gr¹pZ_nqis depicted in Figure 2.1. It is evident from Figure 2.1 that Gr¹pZ_nq does not become disconnected by deletion of any one vertex («- class), whereas, deletion of any two non-adjacent vertices disconnects Gr¹pZ_nq. Hence by Lemma 2.3.12(iii), a minimal separating set of G¹pZ_nq is precisely the union of any two non-adjacent «-classes (in Figure 2.1). By Lemma 2.3.1(ii), we have

|rps| “ pq´1qpr´1q, |rqs| “ pp´1qpr´1q, |rrs| “ pp´1qpq´1q, |rpqs| “ r´1,

|rprs| “q´1 and |rqrs| “p´1. We thus have the following inequalities:

|rps| ą |rqs| ą |rrs|,|rpqs| ą |rprs| ą |rqrs|,|rrs| ą |rprs|. (2.14)