Chapter 3

On (k, d)-multiplicatively indexable

graphs

A (p, q)-graph G is said to be a (k,d)-multiplicatively indexable graph if there exist an

injection f : V (G) → N such that the induced function f× : (E(G)) = {k, k + d, . . . , k +

(q − 1)d} defined by f×(uv) = f(u)f(v) for every uv ∈ E(G). If f(G) = {1, 2, . . . , p}

then G is said to be a (k, d)-strongly-multiplicatively indexable graph. The corresponding

labelings are said to be a (k, d)-multiplicative indexer and a (k, d)-strongly multiplicative

indexer respectively. The first section deals with some properties of (k, d)-multiplicatively

indexable graphs. Then we find a necessary and sufficient condition for the complete

graph C3 to be (k, d) multiplicatively indexable. Moreover we can embed every graph

into a (k, 1)-multiplicatively indexable graph. In the second section we give in some

cases for which strongly (k, d)-multiplicatively indexable graphs does not exists. Also we

show that there exists strongly (k, 1)-multiplicatively indexable graphs for all k > 1 and

strongly (k, 2)-multiplicatively indexable graphs for all k even. Finally we characterize

some particular strongly (k, d)-multiplicatively indexable graphs.

57

Chapter 3 58

3.1 Introduction

B.D. Acharya and S.M. Hegde,[AH91a] defined (p, q)-graph G =

(V, E) multiplicative if there exist an injection f : V (G) → N such

that f×(uv) = f(u)f(v) for every uv ∈ E(G) are all distinct. L.

W. Beineke and S.M. Hegde,[BH01] defined (p, q)-graph G =

(V, E) strongly multiplicative if there exist a bijection f : V (G) →

{1, 2, . . . , p} such that f×(uv) = f(u)f(v) for every uv ∈ E(G) are

all distinct. In this chapter we generalize the multiplicative graphs

into (k, d)-multiplicatively indexable graphs (k,d)-multiplicatively in-

dexable graphs and strongly multiplicative graphs into (k, d)-strongly

multiplicatively indexable graphs. The notion of (k, d)-multiplicative

indexers is closely related to the other known numberings such as

strongly multiplicative labelings and strong indexers, [AH91b] [BH01].

Here we carried out a separate study of (k, d)-multiplicatively index-

able graphs. Most of the theorems and properties given by B.D.

Acharya and S.M. Hegde,[AH90] will be of immense use in our study

here. Also we try to reformulate some results into our terminology.

Such an attempt is given in the following few theorems.

Definition 3.1.1. A (p, q)-graph G is said to be a (k, d) - multiplica-

tively indexable graph if there exist an injection f : V (G) → N such

that the induced function f×(E(G)) = {k, k + d, . . . , k + (q − 1)d}

defined by f×(uv) = f(u)f(v) for every uv ∈ E(G). If f(G) =

{1, 2, . . . , p} then G is said to be a strongly (k, d)- strongly multi-

plicatively indexable graph. The corresponding labelings are called

Chapter 3 59

a (k, d)-multiplicative indexer and a (k, d)-strongly multiplicative in-

dexer respectively.

Figure 3.1 gives two (k, d)-multiplicatively indexable graphs and

Figure 3.2 gives two (k, d)-strongly multiplicatively indexable graphs.

Figure 3.1:

Figure 3.2:

Chapter 3 60

3.2 (k, d)-multiplicatively indexable graphs

.

Theorem 3.2.1. Let G be a (p, q)-graph. Then G is (k, d) - mul-

tiplicatively indexable if and only if it is (m2k,m2d)-multiplicatively

indexable for all non - negative integer m.

Proof. Let G be (k, d) - multiplicatively indexable graph. Hence

there exist a multiplicative indexer f : V (G) → N such that f is

injective and f×(E(G)) = {k, k + d, . . . , k + (q − 1)d}. Now define a

multiplicative labelling f1 : V (G) → N as, f1(ui) = mf(ui) for every

ui ∈ V (G), m > 0. Then f×1 (uiuj) = f1(ui)f1(uj) = mf(ui)mf(uj) =

m2f×(uiuj)

Hence f×(E(G)) = {m2k,m2(k +d), . . . ,m2(k +(q−1)d)}. Therefore

f1 is (m2k,m2d)-multiplicative indexer for m > 0.

Conversely let G is (m2k,m2d)-multiplicatively indexable. Taking

m = 1 we get G is (k, d)-multiplicatively indexable.♦

Corollary 3.2.2. If f is a (k, d)-multiplicative indexer and f1 is a

(m2k,m2d)-multiplicative indexer on G, then∏

f1×(e) = m2q

∏f×(e)∑

f×1 (e) = m2 ∑f×(e) for every e ∈ E(G).

Theorem 3.2.3. Let G = (V, E) be a (p, q)-graph which admits a

(k, d)-multiplicative indexer, f where k and d are not simultaneously

even. Let X = {u ∈ V (G) : f(u) is odd}. Then the number of edges

with both ends in X is

(i)[

q+12

]if k and d are both odd.

Chapter 3 61

(ii)[

q2

]if k is even and d is odd, and

(iii) q if k is odd and d is even.

Proof.

Since k and d are not simultaneously even, the number of odd

numbers in f×(E(G)) is precisely given by (i), (ii) and (iii) in the

respective cases. Further an edge e of G has both ends in X if and

only if f×(e) is odd and hence the result follows.♦

Corollary 3.2.4. If with k is odd and d is even, Then f(u) ≡

1(mod2) for every u ∈ V (G).

Theorem 3.2.5. Let G be a connected (k, k)-multiplicatively index-

able graph. Then for any (k, k)-multiplicative indexer f of G, if k

divides f(u) for every u ∈ V (G)− {v}, then f(v) = 1.

Proof. Since f is a (k, k)-multiplicative indexer of G, then f×(E(G)) =

{jk : 1 ≤ j ≤ q}. Then there exists an edge xy ∈ E(G) with

f×(xy) = k . Assume k be a prime number. Now f×(xy) = f(x)f(y),

which implies either f(x) or f(y) equal to 1, since k divides f(u) for

every u ∈ V (G)− {v}. If k is not a prime say, k = k1k2 where k1 < k

and k2 < k, then there exists no u ∈ V (G) such that f(u) = k1 or k2.

Hence either f(x) or f(y) is to be equal to 1. Since k divides f(u) for

every u ∈ V (G)− {v}, we have x = v so that f(v) = 1.♦

Remark 3.2.6. The converse of the above proposition is not al-

ways true. For example consider the figure 3.3 which is a (3, 3)-

multiplicatively indexable graph. Here f(V (G)) = {1, 2, 3, 4, 5, 6, 9}

Chapter 3 62

and f×(E(G)) = {3, 6, 9, 12, 15, 18}. Note that 1 ∈ f(V (G). However

3 does not divide f(u) for all u ∈ V (G)− {v}.

Figure 3.3:

Remark 3.2.7. However note that every (k, d)-multiplicatively index-

able graph with k prime or square of a prime 1 should be assigned as

a vertex labelling.

Remark 3.2.8. For every (k, d)-multiplicatively indexable graph with

k and d relatively prime 1 ∈ f(V (G))

The following theorem gives a straight forward neessary and suffi-

cient condition for the cycle to be (k, d)-multiplicatively indexable.

Theorem 3.2.9. The cycle C3 is (k, d)-multiplicatively indexable

if and only if there exists three positive integers a, b, c such that

a < b < c and a is the harmonic mean of b and c where k = ab and

d = a(b− c).

Proof. Suppose C3 = (u v w u) is (k, d)-multiplicatively indexable.

Let f : V (C3) → N defined by f(u) = a, f(v) = b and f(w) = c with

a < b < c be a multiplicative indexer of G so that ab = k, bc = k +2d

Chapter 3 63

and ca = k + d. Clearly a = 2bcb+c and d = a(b− c). Conversely if there

exist positive integers a, b, c such that a < b < c and a is the harmonic

mean of b and c then f : V (C3) → N defined by f(u) = a, f(v) = b

and f(w) = c is a (k, d)- multiplicative indexer of G where k = ab and

d = a(b− c).♦

Theorem 3.2.10. The cycle C3 is (k, d)-multiplicatively indexable

for some k, d > 0 then, 2d2 ≡ 0{modk}.

Proof. Since C3 is (k, d)- multiplicatively indexable

then f×(E(C3)) = {k, k + d, k + 2d} = {ab, bc, ca}.

Without loss of generality let ab = k, bc = k + d and ca = k + 2d

then abc2 = (k + d)(k + 2d) so that kc2 = k2 + 3kd + 2d2 hence

2d2 ≡ 0(modk).♦

Theorem 3.2.11. If C3 is (k, d)-multiplicatively indexable for some

k, d > 0 then 1 /∈ f(V (C3)).

Proof. Let V (C3) = {u, v, w} Since C3 is (k, d)-multiplicatively index-

able, there exists f : V (C3) → N and f×(E(C3)) = {k, k + d, k + 2d}.

Let f(u) = 1 and f(v) = b and f(w) = c with b < c. Then

f×(E(C3)) = {b, c, bc}. Since b, c, bc are in arithmetic progression,

we get b = 2c1+c .That is c = b

2−b

Since b > 0 and b 6= 1 we get c < 0 which is a contradiction. Hence

1 /∈ f(V (C3)).♦

Theorem 3.2.12. The cycle C4 is not (k, d)-multiplicatively index-

able for any k, d > 0.

Chapter 3 64

Proof. If possible, assume C4 is (k, d)-multiplicatively indexable for

some k, d > 0

Let V (C4) = {u1, u2, u3, u4} and f : V (C4) → N by

f(u1) = a, f(u2) = x1, f(u3) = b, f(u4) = x2. Then f×(E(C4)) =

{ax1, ax2, bx1, bx2}

Out of the 24 possible numberings of edge values, the only six non-

isomorphic way of numbering of the edges are given below;

ax1, ax2, bx1, bx2

ax1, ax2, bx2, bx1

ax1, bx1, ax2, bx2

ax1, bx1, bx2, ax2

ax1, bx2, ax2, bx1

ax1, bx2, bx1, ax2.

Now consider first way of numbering. Since they are in arithmetic

progression we have

a(x2 − x1) = b(x2 − x1) ⇒ a = b, which is a contradiction to the

fact that f is injective. A similar proof holds for the other way of

numberings also. ♦

The above discussions lead to the following conjecture:

Conjecture 3.2.13. The cycle Cn is (k, d)-multiplicatively index-

able if and only if n ≤ 3.

Theorem 3.2.14. The star K1,b is (k, d)-multiplicatively indexable.

Proof. Let V (K1,b)) = {u0, u1, . . . , ub}. Define f : V (K1,b) → N by

f(u0) = 1 and f(ui) = k + (i − 1)d for 1 ≤ i ≤ b. It can be easily

Chapter 3 65

verified that f is a (k, d) multiplicative indexer of K1,b.♦

Corollary 3.2.15. If k and d are relatively prime, the above (k, d)-

multiplicative indexer is unique in the sense that 1 ∈ f(V (K1,b)) and

1 should be assigned at the central vertex of the star.

it is interesting to see that any (k, d)-multiplicatively indexable

graphs together with a star could be used to generate an infinite as-

cending chain of (k, d)-multiplicatively indexable graphs each of which

contains its predecessor as a subgraph, which leads to the following

theorem.

Theorem 3.2.16. Let G = (V, E) be a (p, q)-graph which admits a

(k, d)-multiplicative indexer f such that there exists exactly one pair of

vertices u, v with f(v) = mf(u) for some non-zero positive integer m.

Consider G1, G2, ..., Gn, n copies of G with (ui, vi) belongs to V (Gi)

such that f(vi) = mf(ui). Let H be the graph obtained by identifying

u2 and v1, u3 and v2, ..., un and vn−1. Then there exists a positive

integer s such that H ∪K1,s is (k, d)-multiplicatively indexable.

Proof. Let G admits a (k, d)-multiplicatively indexaer f .

Let G1, G2, ..., Gn be n copies of G such that

V (G1) = {u1v1, u11, u12, ..., u1(p−2)}

V (G2) = {u2v2, u21, u22, ..., u2(p−2)}

...

...

...

V (Gn) = {unvn, un1, un2, ..., un(p−2)}.

Chapter 3 66

Let uivi ∈ V (Gi) such that f(vi) = mf(ui) and H be the graph ob-

tained by by identifying u2 and v1, u3 and v2, ..., un and vn−1. It is

given that f is a (k, d)-multiplicative indexer on G = G1. Let f(u1) =

a1, f(v1) = ap, f(u11) = a2, f(u12) = a3, f(u1(p−2)) = ap−1 so that

f(v1) = mf(u1). On G2 extend f as f(u2) = f(v1) = mf(u1), f(u3) =

mf(v2) = m2f(u1) and f(u2i) = mf(u1i), 1 ≤ i ≤ p− 2. In general

on Gn, we have f(un) = f(vn−1) = mn−1f(u1), f(vn) = mf(un) and

f(uni) = mf(un−1i), 1 ≤ i ≤ p− 2.

Here all the edge labels of H are the terms of the arithmetic pro-

gression with first term k = minf×(e), e ∈ E(G) and common differ-

ence d. Let the highest edge value of H be the rth term of the above

arithmetic progression,

then k + (r − 1)d = mn−1aimn−1aj = m2n−1aiaj where aiaj =

maxf×(e), e ∈ E(G). Thus we get k + (r− 1)d = m2n−1[k + (q− 1)d]

so that r = m2(n−1)[k+(q−1)d]−kd + 1.

In order to make the graph H to be (k, d)-multiplicatively indexable

we add an isolated vertex w with f(w) = 1. There are qn edges of H.

Hence for each number x with k < x < k + (r − 1)d, x is an element

of the arithmetic progression and x is not an edge label add a vertex

wx such that f(wx) = x.Then the resulting graph obtained by the

union of H and a star K1,s where s = t−qn is a (k, d)-multiplicatively

indexable graph.♦

Chapter 3 67

The following figure 3.4 illustrates the theorem.

Figure 3.4:

We pose the following problem:

Problem 3.2.17. Characterize the class of (k, d)-multiplicatively

indexable graphs.

Theorem 3.2.18. Every graph can be embedded into a (k, 1)-

multiplicatively indexable graph.

Proof. Let G be a graph with n vertices and let V (G) = {u1, u2, . . . , un}.

Let k < p2 < ... < pn where each pi is a prime and let, f(ui) =

pi; 2 ≤ i ≤ n. Now add a vertex v , let f(v) = 1and join v to the

vertices u1 u2, . . . , un so that k, p2, ..., pn are the labels of the edges

vu1, vu2, ..., vun. Now for each integer x with k < x < pn−1pn and x

is not an edge label, add a new vertex vx , let f(vx) = x and join vx

to v. The resulting graph G1 is (k, 1)-multiplicatively indexable and

G is an induced subgraph of G1.♦

Chapter 3 68

Figure 3.5 gives the embedding of a graph with 5 vertices into a

(6, 1)-multiplicatively indexable graph.

Figure 3.5:

3.3 (k, d)-strongly multiplicatively indexable graphs

In this section we establish some classes of graphs which do not admit

a (k, d)-strongly multiplicative indexer for any particular values of k

and d.

Theorem 3.3.1. The complete graph Kn is not (k, d)-strongly multi-

plicatively indexable for n ≥ 3.

Proof. Since Kn for n > 5, f× is not injective it is enough to check for

Kn for n =1, 2, 3, 4, 5. For K1 and K2 the result is trivial. Consider

K3. In this case for any multiplicative labelling f : V (K3) → {1, 2, 3}

Chapter 3 69

the elements of f×(E(G)) = {2, 3, 6} and not in the form {k, k+d, k+

2d}. Hence K3 is not (k, d)-strongly multiplicatively indexable. By a

similar arguments one can show that K4 and K5 are not (k, d)-strongly

multiplicatively indexable.♦

Theorem 3.3.2. There does not exists any (p, q)-graph which is (k, d)-

strongly multiplicatively indexable graphs in the following cases:

(1) For every k odd and d even

(2) For every k even, d odd, and p ≥ d and d does not divide k

(3) For every k odd and d odd and p ≥ d and d does not divide k.

Proof. (1) Since k odd and d even f×E(G)) consists only odd num-

bers. Hence there does not exists a vertex u of G such that f(u) is

even so that f(V (G)) 6= {1, 2, . . . , p} and f is not a (k, d)-strongly

multiplicative indexer of G.

(2) Let k be even, d odd, and p ≥ d and d does not divide k and f

be a (k, d) strongly multiplicative indexer of G. Hence f×(E(G)) =

{k, k+d, . . . , k+(q−1)d}. Also f×(e) ≡ k( mod d) for every e ∈ E(G).

Since p ≥ d and d ∈ f(V (G)) then there exists an edge xy ∈ E(G) with

either, f(x) = d or f(y) = d. In this case f×(xy) ≡ 0(mod d) which

is not possible since d does not divide k and f×(xy) ≡ k(mod d) for

every e ∈ E(G). Hence f is not a (k, d) strongly multiplicative indexer

of G.

(3) The proof of this case follows immediately from case (2).♦

Corollary 3.3.3. For a star K1,p−1 there exists a unique (2, 1)-

strongly multiplicative indexer which assigns 1 to the central vertex

Chapter 3 70

and the numbers 2, 3, . . . , p to the vertices of unit degree.

Theorem 3.3.4. A star K1,n is (k, k)-strongly multiplicatively in-

dexable if n = k − 1. In otherwords the maximum possible edges of a

(k, k)-strongly multiplicatively indexable star is k − 1.

Proof. Since f(K1,n)) = {1, 2, . . . , (n + 1)} and

f×(E(K1,n)) = {k, 2k, . . . , nk}, we should assign the value k to the

central vertex and the numbers 1, 2, ..., n should assign to the pendant

vertices. Hence n = k − 1 and so the maximum possible edges of a

(k, k)-strongly multiplicatively indexable star is k − 1.♦

Theorem 3.3.5. Let k be a positive integer and let n = k2 − 2.

Then the graph G obtained from the star K1,n by subdividing exactly

one edge is (k, k)-strongly multiplicatively indexable.

Proof. Let V (K1,n) = {u, u1, ..., un where n = k2 − 2, degu = n

and let v be the vertex subdividing the edge uu1. Then f : V (G) →

{1, 2, ..., n} defined by

f(u) = k, f(v) = 1 and

f(uj) =

k2 if j = i

j if 1 ≤ j ≤ k − 1

j + 1 if k ≤ j ≤ k2 − 2

is a (k, k)-strongly multiplicative indexer G.♦

Chapter 3 71

The following figure 3.6 illustrates the theorem for k = 3.

Figure 3.6:

Remark 3.3.6. In general the converse of the theorem 3.3.5 is not

true as the figure 3.7 illustrates. Let G be a (4, 4)-strongly multiplica-

tively indexable graph with f(V (G)) = {1, 2, . . . , 8} and f×(E(G)) =

{4, 8, 12, 16, 20, 24, 28}. Here n is not equal to k2 − 2. However , if k

is a prime number, the converse of the above theorem is true which is

our next theorem.

Figure 3.7:

Chapter 3 72

Theorem 3.3.7. Let G be the graph obtained from the star K1,n

by subdividing exactly one edge. If k is a prime number then G is

(k, k)-strongly multiplicatively indexable if and only if n = k2 − 2.

Proof. Let the graph obtained by joining one end vertex of K1,n

to an isolated vertex and assume G is (k, k)-strongly multiplicatively

indexable. Hence f(V (G)) = {1, 2, . . . , (n + 1)} and f×(E(G)) =

{k, 2k, 3k, . . . , nk}. Assume k is a prime number, there exists an edge

e = (uu1) ∈ E(G). Since k is prime, one of f(u) = 1 or f(u1) = 1

and the other is of labelling k. If n = k there exists an edge which is

labelled as k2. By theorem 3.3.4 the possible way of labelling the con-

secutive numbers 2, 3, . . . , k − 1 only to the vertices that are adjacent

to u1 and the value k2 can be assigned to the vertex which is adjacent

to u. Let it be v so that f(v) = k2. Since k2 is in f×(E(G)), we should

assign all the consecutive numbers from k +1 to k2− 1 to the vertices

that are adjacent to u1. Then we have f(V (G)) = {1, 2, . . . , k2} and

f×(E(G)) = {k, 2k, . . . , (k2 − 1)k}. Hence the number of edges of G

is n = k2 − 2.

Converse part follows from the theorem 3.3.5.♦

Theorem 3.3.8. For all positive integer k > 1 there exists a connected

(k, 1)-strongly multiplicatively indexable graph.

Proof. We know that for 2 ≤ m ≤ k, k! + m is divisible by m.

Consider a graph G with k! + 1 vertices and k! edges. Let V (G) =

{u1, u2, . . . , uk!+1} and E(G) = {u1ui : k ≤ i ≤ k! + 1} ∪ {ujuk!+jj

:

2 ≤ j ≤ k − 1}. Define f : V (G) → {1, 2, . . . , k! + 1} as f(ui) =

Chapter 3 73

i, 1 ≤ i ≤ k! + 1 and f induces f× on E(G) as follows: f×(u1ui) =

i, 2 ≤ i ≤ k! + 1 and f×(ujuk!+jj

) = k! + j, k ≤ j ≤ k − 1. Clearly

f×(E(G)) = {k, k + 1, . . . , k! + 1, k! + 2, . . . , k! + k − 1}. That is G is

a connected (k, 1)-strongly multiplicatively indexable graph.♦

Figure 3.8 illustrates the theorem for k = 7.

Figure 3.8:

Chapter 3 74

Theorem 3.3.9. For all positive even integer k there exists a con-

nected (k, 2)-strongly multiplicatively indexable graph.

Proof. Let k be an even number and G be a graph with k!+42 ver-

tices. Let V (G) = {u, v, u1, u2, ..., um, v1, v2, ..., vn} where m = k!−k+42

and n = k−42 so that m + n + 2 = k!+4

2 . Let E(G) = E1(G) ∪

E2(G) ∪ E3(G) where E1(G) = {uui, 1 ≤ i ≤ m}, E2(G) = {vuk2}

and E3(G) = {uj1v1, uj2v2, uj3v3, ..., ujnvn, 1 ≤ ji ≤ m}. Define f :

V (G) → {1, 2, ..., k!+42 } by f(u) = 2, f(v) = 1, f(ui) = i+ k

2 , 1 ≤ i ≤ m

and f(vi) = i + 2, 1 ≤ i ≤ n where n = k−42 . Here we note that

f(u1) = 1 + k2 , f(u2) = 2 + k

2 , in particular f(uk2) = k

2 + k2 = k.

Similarly f(v1) = 3, f(v2) = 4, ..., f(vn) = k2 . Since m + n + 2 =

k!+42 , we have the following relation given by 1 < k!+6

3 < k!+84 <

k!+105 < ... < k!+k

k2

< k!+42 . That is we are selecting certain values from

the set {1, 2, ..., k!+42 }. Let uj1, uj2, ..., ujn be the vertices such that

f(uj1) = k!+63 , f(uj2) = k!+8

4 , ..., f(ujn) = k!+kk2

. Clearly the induced

edge function f× is injective, for if f×(uui) = f(u)f(ui) = 2i + k,

hence f×(E1(G)) = {k + 2, k + 4, ..., k + 2m} where k + 2m = k! + 4.

Also f×(vuk2) = k so that f×(E2(G)) = {k}. Similarly f×(uj1v1) =

k!+63 × 3 = k! + 6, f×(uj2v2) = k!+8

4 × 4 = k! + 8, ..., f×(ujnvn) =

k!+kk2

× k2 = k! + k so that f×(E3(G)) = k! + 6, k! + 8, ..., k! + k. Hence

f×(E(G)) = {k, k + 2, k + 4, ..., k! + 4, k! + 6, k! + 8, ..., k! + k} so that

G is a connected (k, 2)-strongly multiplicatively indexable graph.♦

Chapter 3 75

The following figure 3.9 illustrates the theorem for k = 10.

Figure 3.9:

Theorem 3.3.10. A graph G is (2, 4)-strongly multiplicatively in-

dexable if and only if G ∼= P3

Proof. Let G is (2, 4)-strongly multiplicative. Then f×(e) ≡ 2(mod

4) for every e ∈ E(G). Hence 4 /∈ f×(E(G)) since G is connected

and any edge with one end vertex labelled as 4 is ≡ 0(mod4). So

G ∼= P3.♦

Theorem 3.3.11. (i) The graph G is (2, 4)-strongly multiplicatively

indexable if and only if G ∼= P3.

(ii) The graph G is (2, 3)-strongly multiplicatively indexable if and only

if G ∼= K2.

Chapter 3 76

(iii) The graph G is (2, 10)-strongly multiplicatively indexable if and

only if G ∼= 2K2.

(iv) The graph G is (3, 5)-strongly multiplicatively indexable if and

only if G ∼= 2K2.

Proof. Let G be (2, 4)-strongly multiplicative indexer. Then f×(e) ≡

2(mod4) for every e ∈ E(G) so that 4 /∈ f(V (G)). Thus |V (G)| = 3

and hence G ∼= P3.Conversely if G = P3 = (v1, v2, v3)then f defined

by f(v1) = 1, f(v2) = 2, f(v3) = 3 is a (2, 4)-strongly multiplicative

indexer of G.

The proof of (ii), (iii) and (iv) are similar.♦

Chapter 3 77

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