some results on analytic odd mean labeling of graph

BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTEISSN (p) 2303-4874, ISSN (o) 2303-4955www.imvibl.org /JOURNALS / BULLETINVol. 9(2019), 487-499

DOI: 10.7251/BIMVI1903487J

FormerBULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA LUKA

ISSN 0354-5792 (o), ISSN 1986-521X (p)

SOME RESULTS ON

ANALYTIC ODD MEAN LABELING OF GRAPH

P. Jeyanthi, R.Gomathi, and Gee-Choon Lau

Abstract. Let G = (V,E) be a graph with p vertices and q edges. Agraph G is analytic odd mean if there exist an injective function f : V →{0, 1, 3, 5 . . . , 2q − 1} with an induce edge labeling f∗ : E → Z such that for

each edge uv with f(u) < f(v),

f∗(uv) =

⌈f(v)2−(f(u)+1)2

2

⌉, if f(u) 6= 0⌈

f(v)2

2

⌉, if f(u) = 0.

is injective. We say that f is an analytic odd mean labeling of G. In this paper

we prove that quadrilateral snake Q(n), double quadrilateral snake DQ(n),

coconut tree, fire cracker graph, splitting graph spl(G), Pn(1, 2, 3, . . . , n), andthe graph Ck � K̄n are analytic odd mean graphs.

1. Introduction

Throughout this paper by a graph we mean a finite, simple and undirectedone.The vertex set and the edge set of a graph G are denoted by V (G) and E(G)respectively.Terms and notations not defined here are used in the sense of Harary[1]. A graph labeling is an assignment of integers to the vertices or edges or both,subject to certain conditions. There are several types of labeling. An excellent sur-vey of graph labeling is available in [5]. The concept of analytic mean labeling wasintroduced in [6]. A (p, q) graph G(V,E) is said to be an analytic mean graph if it ispossible to label the vertices in V with distinct elements from 0, 1, 2, . . . , p−1 in sucha way that when each edge e = uv is labelled with f∗(uv) =

∣∣(f(u))2 − f(v)2∣∣ /2 if∣∣(f(u))2 − f(v)2

∣∣ is even and (∣∣(f(u))2 − f(v)2

∣∣ + 1)/2 if∣∣(f(u))2 − f(v)2

∣∣ is odd

2010 Mathematics Subject Classification. 05C78.Key words and phrases. mean labeling, analytic mean labeling, analytic odd mean labeling,

analytic odd mean graph.

487

488 P. JEYANTHI, R.GOMATHI, AND GEE-CHOON LAU

and the edge labels are distinct. Motivated by the results in [6], we introduced anew mean labeling called analytic odd mean labeling in [2]. We proved that cycleCn, path Pn, n-bistar, comb Pn � K1, graph Ln � K1, wheel graph Wn, flowergraph Fln, some splitting graphs, multiple of graphs, the square graph of Pn, Cn,Bn,n, H-graph and H � mK1 fan Fn, double fan D(Fn), double wheel D(Wn),closed helm CHn, total graph of cycle T (Cn), total graph of path T (Pn), armedcrown CnΘPm, generalized peterson graph GP (n, 2) are analytic odd mean graphsin [2], [3] and [4].

We use the following definitions in the subsequent section.

Definition 1.1. Let G(V,E) be a graph with p vertices and q edges. Wesay G is an analytic odd mean graph if there exist an injective function f from thevertex set to 0, 1, 3, 5, . . . , 2q − 1 in such a way that when each edge e = uv suchthat f(u) < f(v) is labeled with f∗(uv) =

⌈f(v)2 − (f(u) + 1)2/2

⌉if f(u) 6= 0 and

f∗(uv) =⌈f(v)2/2

⌉if f(u) = 0 all edge labels are odd and distinct. In this case f

is called an analytic odd mean labeling.

Definition 1.2. LetQ(n) be the quadrilateral snake obtained from the pathv1, v2, . . . , vn+1 by joining vi and vi+1 to the new vertices ui and wi for 1 6 i 6 n.That is, every edge of a path is replaced by a cycle C4.

Definition 1.3. Let DQ(n) be the quadrilateral snake obtained from thepath v1, v2, v3, . . . , vn+1. The double quadrilateral snake DQ(n) is obtained fromQ(n) by adding the vertices s1, s2, s3, . . . , sn; t1, t2, t3, . . . , tn and the edges visi,tivi+1, siti for 1 6 i 6 n.

Definition 1.4. The splitting graph spl(G) is obtained by adding a newvertex v′ corresponding to each vertex v of G such that N(v) = N(v′).

Definition 1.5. The graph Pn(1, 2, 3, . . . , n) is a graph obtained from a pathof vertices v1, v2, v3, . . . , vn by joining i pendent vertices at each ith vertex 1 6 i 6 n.The pendent vertices are labeled as uij for 1 6 i 6 n and 1 6 j 6 i.

Definition 1.6. The fire cracker is constructed as follows: Let a0, a1, a2, . . . ,ak−1 be the vertices of the path Pk and bj be the vertex adjacent to aj for 1 6 j 6 k.Let bj1, bj2, bj3, . . . , bjn be the pendent vertices adjacent to bj for 1 6 j 6 k.

Definition 1.7. The coconut tree is having the vertices v0, v1, v2, . . . , vi ofa path (i > 1) and the pendent vertices vi+1, vi+2, vi+3, . . . , vi+n, being adjacentwith v0.

Definition 1.8. The corona G1 �G2 of two graphs G1 and G2 is defined asthe graph G obtained by taking one copy of G1 (which has p vertices) and p copiesof G2 and then joining the ith vertex of G1 to every vertex in the the ith copy ofG2 for 1 6 i 6 p.

2. Main Results

In this section we prove that quadrilateral snake Q(n), double quadrilateralsnake DQ(n), coconut tree, fire cracker graph and some special star graphs, splittinggraph spl(G), Pn(1, 2, 3, ·· ·n) and the graph Ck�K̄n are analytic odd mean graphs.

SOME RESULTS ON ANALYTIC ODD MEAN LABELING OF GRAPH 489

Theorem 2.1. The splitting graph spl(Pn) is an analytic odd mean graph.

Proof. Let V (G) = {vi, v′i : 0 6 i 6 n− 1} and

E(G) = {vi−1vi, v′i−1vi, vi−1v′i : 1 6 i 6 n− 1}.

Now |V (G)| = 2n and |E(G)| = 3n − 3. We define an injective map f : V (G) →{0, 1, 3, 5, . . . , 6n− 7} by

f(v0) = 0, f(vi) = 2i− 1 for 1 6 i 6 n− 1 and f(v′i−1) = 2n− 3 + 2i for 1 6 i 6 n.

Let f∗ be the induced edge labeling of f . The induced edge labels are as follows:f∗(vi−1vi) = 2i− 1 for 1 6 i 6 n− 1f∗(v′i−1vi) = 2n(n− 3) + 2i(2n− 3) + 5 for 1 6 i 6 n− 1f∗(viv

′i+1) = 2n(n + 1) + 2i(2n + 1) + 1 for 1 6 i 6 n− 2 and

f∗(v0v′1) = 2n2 + 2n + 1.

Clearly all the edge labels are odd. We observe that the edge labels vi−1vi increaseby 2 from 1 to 2n-3 as i increases from 0 to n-1. Also the edge labels of v′i−1viincrease by 4n-6 from 2n2−2n−1 to 6n2−16n+11 as i increases from 1 to n-1 andthe edge labels of viv

′i+1 increase by 4n+2 from 2n2 + 2n + 1 to 6n2 − 4n− 3 as i

increases from 1 to n-2. Moreover, f∗(v′i−1vi) 6= f∗(viv′i+1) so all the edge labels are

distinct. Hence the splitting graph spl(Pn) admits an analytic odd mean labeling.An analytic odd mean labeling of splitting graph spl(P6) is shown in Figure 1.

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t t t t t t

0 1 3 5 7 9

11 13 191715 21

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Figure 1

Theorem 2.2. The splitting graph spl(Cn) is an analytic odd mean graph.

Proof. Let V (G) = {vi, v′i : 0 6 i 6 n− 1} and

E(G) = {vi−1vi : 1 6 i 6 n− 1} ∪ {v′ivi+1 : 1 6 i 6 n− 2}∪{v′ivi−1 : 1 6 i 6 n− 1}{v′n−1v0, v′0vn−1}.

Now |V (G)| = 2n and |E(G)| = 3n. We define an injective map f : V (G) →{0, 1, 3, 5, . . . , 6n− 1} by

f(v0) = 0 and f(vi) = 2i− 1 for 1 6 i 6 n− 1,

f(v′i) = 2n + 2i− 1 for 0 6 i 6 n− 1.


Let f∗ be the induced edge labeling of f . The induced edge labels are as follows:f∗(vi−1vi) = 2i− 1 for 1 6 i 6 n− 1f∗(v′ivi+1) = 2n(n− 1) + 2i(2n− 3)− 1 for 1 6 i 6 n− 2f∗(v′ivi−1) = 2n(n− 1) + 2i(2n + 1)− 1 for 1 6 i 6 n− 1f∗(v′n−1v0) = 8n2 − 12n + 5 andf∗(vn−1v

′0) = 2n− 1.

It is easy to show that the edge labels are odd and distinct. Hence the splittinggraph spl(Cn) admits an analytic odd mean labeling. An analytic odd mean label-ing of splitting graph spl(C8) is shown in Figure 2.

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Figure 2

Theorem 2.3. The splitting graph spl(Km,n) for any integer n > m > 1 is ananalytic odd mean graph.

Proof. LetV (G) = {ui, u

′i, un+j , u

′n+j : 0 6 i 6 n− 1 and 1 6 j 6 m} and

E(G) = {uiun+j , uiu′n+j , u

′iun+j : 0 6 i 6 n− 1 and 1 6 j 6 m}

be the vertex set and edge set of the graph G = spl(Km,n). Now |V (G)| = 2(m+n)and |E(G)| = 3mn. We define an injective map f : V (G)→ {0, 1, 3, 5, . . . , 6mn−1}by


f(u0) = 0,f(ui) = 2i− 1 for 1 6 i 6 n− 1 andf(u′i) = 2n + 2i− 1 for 0 6 i 6 n− 1.

We label m vertices un+j by 6mn-1,6mn-3,6mn-5,6mn-2m+1. That is f(un+j) =6mn− 2j + 1 for 1 6 j 6 m.

Also we label m vertices u′n+j by 6mn-2m-1,6mn-2m-3,6mn-2m-5,6mn-4m+1.That is f(u′n+j) = 6mn−2m−2j + 1 for 1 6 j 6 m. Then the induced edge labelsare as follows:

f∗(un+jui) = [(6mn− 2j + 1)2 − (2i)2 + 1]2 = 6mn(3mn + 1) + 1− 2j(6mn +1) + 2j2 − 2i2 for 0 6 i 6 n− 1 and 1 6 j 6 m.

f∗(u′n+jui) = [(6mn− 2m− 2j + 1)2 − (2i)2 + 1]2 = [(6mn− 2m + 1)2 + 1]÷2− 2j(6mn− 2m + 1) + 2j2 − 2i2) for 0 6 i 6 n− 1&1 6 j 6 m.

f∗(un+ju′i) = [(6mn−2j+1)2−(2n+2i)2+1]2 = 6mn(3mn+1)+1−2j(6mn+

1) + 2j2 − 2i2 − 2n2 − 4ni for 0 6 i 6 n− 1 and 1 6 j 6 m.

We observe that the vertices un+1 to un+m and u0, u1, . . . , un−1, u′0, u′1, . . . , u

′n−1

induce Km,2n whereas the vertices u0 to un−1 and un+1, un+2, . . . , un+m, u′n+1,u′n+2

, . . . , u′n+m induced a Kn,2m. By an argument similar to that for Km,n, we can seethe labeling is analytic odd mean labeling . The analytic odd mean labeling ofsplitting graph spl(K3,4) is shown in Figure 3.

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un+3

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Figure 3

Theorem 2.4. The graph Pn(1, 2, 3, . . . , n) is an analytic odd mean graph.

Proof. Let G = Pn(1, 2, 3, . . . n). Let

V (G) = {vi, vji : 0 6 i 6 n− 1 and 1 6 j 6 i + 1} and


E(G) = {vi−1vi : 1 6 i 6 n− 1} ∪ {vivji : 0 6 i 6 n− 1 and 1 6 j 6 i + 1}.Now |V (G)| = n + n(n + 1)÷ 2 and |E(G)| = n− 1 + n(n + 1)÷ 2. We define aninjective map f : V (G)→ {0, 1, 3, 5, . . . , n2 + 3n− 3} by

f(v0) = 0, f(vi) = 2i− 1 for 1 6 i 6 n− 1, f(v10) = 2n− 1 and

f(vji ) = 2n− 3 + 2(i + j) +∑i−1

k=1 2(i− k) for 1 6 i 6 n− 1 and 1 6 j 6 i + 1.

Let f∗ be the induced edge labeling of f . The induced edge labels are as follows:

f∗(vi−1vi) = 2i− 1 for 1 6 i 6 n− 1

f∗(v0v10) = 2n2 − 2n + 1 and

f∗(vivji ) = 2(n2−i2)−6n+5+2[(i+j)+

∑i−1k=1(i−k)][(i+j)+

∑i−1k=1(i−k)+2n−3]

for 1 6 j 6 m and 1 6 i 6 n− 1.

Clearly the edge labels are odd and We observe that the edge labels are increase by2 as i increases from 0 to n-1. For fix i, the difference of f∗(viv

ji ) and f∗(viv

j+1i )

is 4[i + j +∑i−1

k=1(i − k) + n − 1]. So all the edge labels are distinct. Hence thegraph G admits an analytic odd mean labeling. An analytic odd mean labeling ofP5(1, 2, 3, . . . , 5) is shown in Figure 4.

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s s s s s s

s

s s s s s s s s s

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v0 v1 v2 v3 v5

v10 v11 v21 v12 v22 v32 v13 v23 v33 v43 v15 v25 v35 v45 v55

0 1 3 5 7

9 11 13 15 17 19 21 23 2527 31 33 35 37 39

Figure 4

Theorem 2.5. The quadrilateral snake Q(n) is an analytic odd mean graph.

Proof. Let v1, v2, v3, . . . , vn+1;u1, u2, u3, . . . , un;w1, w2, w3, . . . , wn be the ver-tices of V (Qn) and E(Wn) = {vivi+1, viui, uiwi, wivi+1 : 1 6 i 6 n}. Let G = Q(n)and hence |V (G)| = 3n + 1 and |E(G)| = 4n. We define f on the vertex set of Qn

as follows :f(vi) = 2i− 1 for 1 6 i 6 n + 1f(ui) = 2n + 1 + 2i for 1 6 i 6 nf(wi) = 4n + 1 + 2i for 1 6 i 6 n.

Let f∗ be the induced edge labeling of f . The induced edge labels are as follows :

f∗(vivi+1) = 2i + 1 for 1 6 i 6 n

f∗(viui) = 2n(n + 1) + 2i(2n + 1) + 1 for 1 6 i 6 n

f∗(uiwi) = 6n2 + 2i(2n− 1)− 1 for 1 6 i 6 n and

f∗(wivi+1) = 4n(2n + 1) + 2i(4n− 1)− 1 for 1 6 i 6 n.

Clearly the edge labels are odd and we observe that the edge labels of path increaseby 2 , the edge labels of viui increase by 4n+2, the edge labels of uiwi increase by4n-2 and the edge labels of wivi+1 increase by 8n-2 as i increases. So all the edge


labels are distinct. Therefore f is an analytic odd mean labeling and hence Q(n)is an analytic odd mean graph. An analytic odd mean labeling of Q(5) is shown inFigure 5.

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v1 v2 v3 v4 v5 v6

13 23 15 25 17 27 19 29 21 31

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Figure 5

Theorem 2.6. The double quadrilateral snake D(Q(n)) is an analytic odd meangraph.

Proof. Let

v1, v2, v3, . . . , vn+1;u1, u2, u3, . . . , un;w1, w2, w3, . . . , wn; s1, s2, s3, . . . , sn; t1, t2, t3, . . . , tn

be the vertices of V (Qn) .Let G = D(Q(n)). Hence |V (G)| = 5n + 1 and |E(G)| =7n. We define an injective map f : V (G)→ {0, 1, 3, 5, . . . , 14n− 1} by

f(vi) = 2i− 1 for 1 6 i 6 n + 1f(ui) = 2n + 1 + 2i for 1 6 i 6 nf(wi) = 4n + 1 + 2i for 1 6 i 6 nf(si) = 6n + 1 + 2i for 1 6 i 6 n andf(ti) = 8n + 1 + 2i for 1 6 i 6 n.

Let f∗ be the induced edge labeling of f . Then the induced edge labels are asfollows:

f∗(vivi+1) = 2i + 1 for 1 6 i 6 n

f∗(viui) = 2n(n + 1) + 2i(2n + 1) + 1 for 1 6 i 6 n

f∗(uiwi) = 6n2 + 2i(n− 1)− 1 for 1 6 i 6 n

f∗(uisi) = 6n(3n + 1) + 2i(6n + 1) + 1 for 1 6 i 6 n

f∗(wivi+1) = 4n(2n + 1) + 2i(4n− 1)− 1 for 1 6 i 6 n− 1

f∗(tivi+1) = 8n(4n + 1) + 2i(8n− 1)− 1 for 1 6 i 6 n and

f∗(siti) = 2n(7n− 2) + 2i(2n− 1)− 1 for 1 6 i 6 n.

We observe that the edge labels are odd and distinct. Therefore f is an analyticodd mean labeling and hence D(Q(n)) is an analytic odd mean graph. An analyticodd mean labeling of D(Q(4)) is shown in Figure 6.

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Theorem 2.7. The graph Ck � K̄n is an analytic odd mean graph.

Proof. Let G = Ck � K̄n. Let

V (G) = {vi, vji : 0 6 i 6 k − 1 and 1 6 j 6 n}

and

E(G) = {vivi+1 : 0 6 i 6 k−2}∪{vivji : 0 6 i 6 k−1 and 1 6 j 6 n}∪{vk−1v0}.

Then there are k(n+1) vertices and deg(vi) = n + 2 for 0 6 i 6 k− 1. We define fon the vertex set as follows :

f(v0) = 0, f(vi) = 2i− 1 for 1 6 i 6 k − 1and

f(vji ) = 2k − 3 + 2j + 2ni for 0 6 i 6 k − 1 and 1 6 j 6 n.

Let f∗ be the induced edge labeling of f.Then the induced edge labels are as follows:

f∗(vivi+1 = 2i− 1 for 1 6 i 6 k − 2

f∗(vk−1v0) = 2k2 − 6k + 5

f∗(vivji ) = 2k(k−3)+2(2k+ni−3)(ni+j)+2nij+5+2(j2−i2) for 1 6 i 6 k−1

and 1 6 j 6 n and

f∗(v0vj0) = 2(k + j)(k + j − 3) + 5 for 1 6 j 6 n.

It can be easily verified that f is an analytic odd mean labeling and hence graphCk� K̄n is an analytic odd mean graph. An analytic odd mean labeling of C5� K̄3

is shown in Figure 7.

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Figure 7

Theorem 2.8. Let S1, S2, S3, . . . Sk be the disjoint copies of the k-star K1,k withvertex set V (Si) = {vi, vi,r : 1 6 r 6 k} and the edge set E(Si) = {vivi,r : 1 6r 6 k} for 1 6 i 6 k and G be the graph obtained by joining a new vertex v withv1,1, v2,1, v3,1, . . . vk,1. Then G is an analytic odd mean graph.

Proof. Let V (G) = {v, vi, vi,r : 1 6 i, r 6 k} and E(G) = {vvi,1, vivi,r : 1 6i, r 6 k}. We define an injective map f : V (G)→ {0, 1, 3, 5, . . . , 2k2 + 2k − 1} by

f(v) = 0, f(vi) = 2i− 1 for 1 6 i 6 k andf(vi,r) = 2ki + 2r − 1 for 1 6 i, r 6 k.

Then the induced edge labeling of f are

f∗(vvi,1) = 2k2i2 + 2ki + 1 for 1 6 i 6 k

f∗(vivi,r) = 2i(i(k2 − 1) + kr) + 2(r − 1)(ki + r) + 1 for 1 6 i, r 6 k.

Clearly the edge labels are odd and distinct. Hence the graph admits G an analyticodd mean labeling. An analytic odd mean labelling of the above graph G withk = 6 is shown in Figure 8.

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Theorem 2.9. Let S1, S2, S3, . . . , Sk+1 be the disjoint copies of the k-star K1,k

with vertex set V (Si) = {vi, vri : 1 6 r 6 k} and the edge set E(Si) = {vivri : 1 6r 6 k} for 1 6 i 6 k + 1. Let G be the graph obtained by joining a new vertex v tothe centre of each k + 1 stars. Then G is an analytic odd mean graph.

Proof. Let V (G) = {v, vi, vri : 1 6 i 6 k + 1 and 1 6 r 6 k} and E(G) ={vvi, vivri : 1 6 i 6 k + 1 and 1 6 r 6 k}. We define an injective map f : V (G)→{0, 1, 3, 5, . . . , 2k2 + 4k + 1} by

f(v) = 0, f(vi) = 2i− 1 for 1 6 i 6 k + 1 andf(vri ) = 2ki + 2r + 1 for 1 6 i 6 k + 1 and 1 6 r 6 k.

Then the induced edge labeling of f are

f∗(vvi) = 2i− 1 for 1 6 i 6 k + 1 andf∗(viv

ri ) = 2i(i(k2 − 1) + kr) + 2(r − 1)(ki + r) + 1 for 1 6 i 6 k + 1 and

1 6 r 6 k.

Clearly the edge labels are odd and distinct. Hence the graph admits G an analyticodd mean labeling. An analytic odd mean labeling of the graph G with k = 4 isshown in Figure 9.

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Theorem 2.10. The fire cracker is an analytic odd mean graph.

Proof. LetV (G) = {ai, bj , bkj : 0 6 i 6 n− 1, 1 6 j 6 n and 1 6 k 6 m} and

E(G) = {aiai+1 : 0 6 i 6 n−2}∪{aibj : 0 6 i 6 n−2 1 6 j 6 n}∪{bjbkj : 1 6j 6 n and 1 6 k 6 m}.We define an injective map f : V (G)→ {0, 1, 3, 5, . . . , 2n + nm− 1} by

f(a0) = 0, f(ai) = 2i− 1 for 1 6 i 6 n− 1f(bj) = 2n− 3 + 2j for 1 6 j 6 n andf(bkj ) = 4n− 3 + 2m(j − 1) + 2k for 1 6 j 6 n and 1 6 k 6 m.

Let f∗ be the induced edge labeling of f .Then the induced edge labels are as follow:

f∗(ai−1ai) = 2i− 1 for 1 6 i 6 n

f∗(a0b1) = 2n2 − 2n + 1

f∗(aibj) = 2(n − 1)(n + i) + 2ni + 1 for 0 6 i < j 6 n f∗(bjbkj ) = 2(3n2 +

k2 − j2)− 4n(j + 2) + 2k(4n− 3) + 2m(j − 1)[m(j − 1) + 4n− 3 + 2k] + 4j + 3 for1 6 j 6 n and 1 6 k 6 m.

Clearly the edge labels are odd and distinct. Hence the fire cracker admits analyticodd mean labeling. An analytic odd mean labeling of fire cracker for n,m = 3 is


shown in Figure 10.

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189

9

33

27

25

23

215

263

3155

13

15

13

11

43

67

95

Figure 10

Theorem 2.11. The coconut tree G is an analytic odd mean graph.

Proof. LetV (G) = {vi, vn+j : 1 6 i 6 n and 1 6 j 6 m} andE(G) = {vi−1vi : 1 6 i 6 n− 1} ∪ {v0vn+j : 1 6 j 6 m}.

Now |V (G)| = n + m and |E(G)| = m + n − 1. We define an injective mapf : V (G)→ {0, 1, 3, 5, . . . , 2(m + n)− 3} by

f(v0) = 0, f(vi) = 2i− 1 for 1 6 i 6 n− 1 andf(vn+j) = 2n− 3 + 2j for 1 6 j 6 m.

Let f∗ be the induced edge labeling of f . The induced edge labels are as follows:

f∗(vi−1vi) = 2i− 1 for 1 6 i 6 n− 1 and

f∗(v0vn+j) = 2n(n− 3) + 2j(2n− 3) + 2j2 + 5 for 1 6 j 6 m.

Clearly the edge labels are odd and we observe that the edge labels of path increaseby 2 as i increases and the pendent edge labels increase by

4n, 4(n + 1), 4(n + 2), 4(n + 3), . . .


as j increases from 1 to m. So all the edge labels are distinct. Hence the coconuttree admits an analytic odd mean labeling. An analytic odd mean labeling of co-conut tree with n = 6,m = 9 is shown in Figure 11.

�

u

uu

u

uu

u

uu u

u u u

uu0

1

3

5

7

9

1

3

5

7

9

11

13

15

1719

21

23

25

2761

85

113

145181

221

313

365

265

Figure 11

References

[1] F. Harary. Graph Theory Addison-Wesley, Reading, Massachusetts, 1972.

[2] P. Jeyanthi, R.Gomathi and Gee-Choon Lau. Analytic odd mean labeling of square andH-graphs. International J. Math. Combinatorics, Special issue 1(2018), 61-67.

[3] P. Jeyanthi, R. Gomathi and Gee-Choon Lau. Analytic odd mean labeling of some stan-

dard graphs, In: Proceedings of the International Conference on Algebra and DiscreteMathematics-2018 (ICADM-2018), (pp. 62-68). Madurai, India, 2018.


[4] P. Jeyanthi, R. Gomathi and Gee-Choon Lau. Analytic odd mean labeling of some graphs.

Palestine Journal of Mathematics, to appear.

[5] J. A. Gallian. A Dynamic Survey of Graph Labeling. The Electronic Journal of Combina-torics, (2016), # DS6.

[6] T. Tharmaraj and P. B. Sarasija. Analytic mean labelled graphs. International Journal ofMathematical Archive, 5(6)(2014), 136–146.

Receibed by editors 18.09.2018; Revised version 27.03.2019; Available online 08.04.2019.

Research Centre, Department of Mathematics, Govindammal Aditanar College for

Women, Tiruchendur 628215, Tamilnadu, India.E-mail address: [email protected]

Research Scholar,Reg.No.:11825, Research Centre, Govindammal Aditanar Col-lege for Women, Tiruchendur-628 215, Affliated to Manonmaniam Sundaranar Univer-

sity, Abishekapatti, Tirunelveli-627 012, Tamilnadu, India.

E-mail address: [email protected]

Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA

(Segamat Campus), 85009, Johor, Malaysia., Abishekapatti, Tirunelveli-627 012, Tamil-

nadu, India.E-mail address: [email protected]

some results on analytic odd mean labeling of graph

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