View
13
Download
0
Category
Preview:
Citation preview
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
References
[AG96] S. Arumugam and K.A. Germina. On indexable graphs. Dis-
crete Mathematics, (161):285–289, 1996.
[AH90] B.D. Acharya and S. M. Hegde. Arithmetic graphs. Journal
of Graph Theory, 14:275–299, 1990.
[AH91a] B.D. Acharya and S. M. Hegde. On certain vertex valuations
of graphs. Indian J. Pure Appl Math., 22:553–560, 1991.
[AH91b] B.D. Acharya and S.M. Hegde. Strongly indexable graphs.
Discrete Mathematics, (93):123–129, 1991.
[BH01] L.W. Beineke and S. M. Hegde. Strongly multiplicative
graphs. Discussions Mathematicac, Graph Theory., 21:63–
75, 2001.
[Bur06] David M. Burton, editor. Elementary Number Theory. TATA
McGRAW-HILL,, 2006.
[Gal05] J.A. Gallian. A dynamic survey of graph labelling. The Elec-
tronic Journal of Combinatorics(DS6), pages 1–148, 2005.
[Gol80] Martin Charles Goloumbic. Algorithmic Graph Theory and
Perfect Graphs. Academic Press Inc., New York, 1980.
[Har72] Frank Harary. Graph Theory. Addision Wesley, Mas-
sachusetts, 1972.
Recommended