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International J.Math. Combin. Vol.2(2020), 109-117
Triangular Difference Mean Graphs
P. Jeyanthi1, M. Selvi2and D. Ramya3
1.Research Centre, Department of Mathematics, Govindammal Aditanar College for Women
Tiruchendur-628 215, Tamil Nadu, India
2.Department of Mathematics, Dr. Sivanthi Aditanar College of Engineering
Tiruchendur-628 215, Tamil Nadu, India
3.Department of Mathematics, Government Arts College (Autonomous)
Salem-7, Tamil Nadu, India
E-mail: jeyajeyanthi@rediffmail.com, selvm80@yahoo.in, aymar padma@yahoo.co.in
Abstract:In this paper, we define a new labeling namely triangular difference mean
labeling and investigate triangular difference mean behaviours of some standard graphs. A
triangular difference mean labeling of a graph G= (p, q) is an injection f:V−→ Z+,where
Z+is a set of positive integers such that for each edge e=uv, the edge labels are defined as
f∗(e) = |f(u)−f(v)|
2
such that the values of the edges are the first qtriangular numbers. A graph that admits a
triangular difference mean labeling is called a triangular difference mean graph.
Key Words:Mean labeling, triangular difference mean labeling, Smarandachely k-
triangular labeling, triangular difference mean graph.
AMS(2010):05C78.
§1.Introduction
By a graph, we mean a finite, simple and undirected one. The vertex set and the edge set of a
graph Gare denoted by V(G) and E(G) respectively. Terms and notations not defined here are
used in the sense of Harary [2] and for number theory we follow Burton[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 graph labeling and an excellent survey on graph labeling can be found in [3]. The
notion of triangular mean labeling was due to Seenivasan et al. [7]. Let G= (V , E) be a graph
with pvertices and qedges. Consider an injection f:V(G)−→ {0,1,2,· · · , Tq}, where Tqis the
qth triangular number. Define f∗:E(G)−→ {1,3,· · · , Tq}such that f∗(e) = lf(u)+f(v)
2mfor
all edges e=uv. If f∗(E(G)) is a sequence of consecutive triangular numbers T1, T2,· · · , Tq,
then the function fis said to be triangular mean labeling. Generally, If there are only k
consecutive triangular numbers Ti, Ti+1,· · · , Ti+k−1with k≤qin f∗(E(G)), such a fis called
a Smarandachely k-triangular labeling. A graph that admits a triangular mean labeling or
Smarandachely k-triangular labeling is called a triangular mean graph or a Smarandachely
1Received November 21, 2019, Accepted June 13, 2020.
110 P. Jeyanthi, M. Selvi and D. Ramya
k-triangular mean graph.
Murugan et al.[4] introduced skolem difference mean labeling and some standard results
on skolem difference mean labeling were proved in [5] and [6]. A graph G= (V, E) with p
vertices and qedges is said to have skolem difference mean labeling if it is possible to label
the vertices x∈Vwith distinct elements f(x) from {1,2,3,· · · , p +q}in such a way that for
each edge e=uv, let f∗(e) = l|f(u)−f(v)|
2mand the resulting labels of the edges are distinct
and are 1,2,3,· · · , q. A graph that admits a skolem difference mean labeling is called a skolem
difference mean graph.
Motivated by the concepts in [7] and [4], we define a new labeling namely triangular
difference mean labeling. A triangular difference mean labeling of a graph G= (p, q) is an
injection f:V−→ Z+, where Z+is a set of positive integers such that for each edge e=uv,
the edge labels are defined as f∗(e) = l|f(u)−f(v)|
2msuch that the values of the edges are the
first qtriangular numbers. A graph that admits a triangular difference mean labeling is called
a triangular difference mean graph. We use the following definitions in the subsequent sequel.
Definition 1.1A vertex of degree one is called a pendant vertex and a pendant edge is an edge
incident with a pendant vertex. The corona G1G2of the graphs G1and G2is obtained by
taking one copy of G1(with pvertices) and pcopies of G2and then join the ith vertex of G1
to every vertex of the ith copy of G2.
Definition 1.2The bistar Bm,n is a graph obtained from K2by joining mpendant edges to
one end of K2and npendant edges to the other end of K2.
Definition 1.3The graph Cn@Pmis obtained by identifying one pendant vertex of the path
Pmto a vertex of the cycle Cn.
Definition 1.4A triangular number is a number obtained by adding all positive integers less
than or equal to a given positive integer n. If the nth triangular number is denoted by Tn, then
Tn=1
2n(n+ 1).
§2.Triangular Difference Mean Graphs
In this section, we establish that path Pn(n≥1), K1,n(n≥1), PnK1(n≥2), Bm,n (m≥
1, n ≥1), T(n, m), S(n, n, · · · , n
| {z }
m times
), Cn(n > 3) and Cn@Pm(n≥4, m ≥2) admit triangular
difference mean labeling . Further, we prove that C3is not a triangular difference mean graph.
Theorem 2.1Any path Pn(n≥1) is a triangular difference mean graph.
Proof Let v1, v2,· · · , vnbe the vertices of the path Pn. Then E(Pn) = {ei=vivi+1 : 1 ≤
i≤n−1}. Define f:V(Pn)−→ Z+as follows:
f(v1) = 1 and f(vi) = 2(T1+T2+· · · +Ti−1) + 1 for 2 ≤i≤n.
For the vertex label f, the induced edge label f∗is as follows:
Triangular Difference Mean Graphs 111
f∗(ei) = Tifor 1 ≤i≤n−1. Hence Pnis a triangular difference mean graph.
The triangular difference mean labeling of P5is given in Figure 1.
rr
r rr
1 412193
Figure 1
Theorem 2.2The star graph K1,n(n≥1) admits triangular difference mean labeling.
Proof Let vbe the apex vertex and v1, v2,· · · , vnbe the pendant vertices of the star K1,n.
Then E(K1,n) = {vvi: 1 ≤i≤n}. Define f:V(K1,n)−→ Z+as follows:
f(v)=1, f (vi)=2Ti+ 1 for 1 ≤i≤n.
For the vertex label f, the induced edge label f∗is as follows:
f∗(vvi) = Tifor 1 ≤i≤n.
Then the induced edge labels are the triangular numbers T1, T2,· · · , Tn. Hence K1,n is a
triangular difference mean graph.
The triangular difference mean labeling of K1,8is shown in Figure 2.
r
r
r
r
r
r
r
r
r
3
1
7
13
21
31
43
57
73
Figure 2
Theorem 2.3The comb graph PnK1(n≥2) admits triangular difference mean labeling.
Proof Let v1, v2,· · · , vnbe the vertices of the path Pnand u1, u2,· · · , unbe the pendant
vertices adjacent to v1, v2,· · · , vnrespectively. Then E(PnK1) = {ei=vivi+1, e0
j=ujvj:
1≤i≤n−1,1≤j≤n}. Define f:V(PnK1)−→ Z+as follows:
f(v1)=1, f (vi) = 2(T1+T2+· · · +Ti−1) + 1 for 2 ≤i≤n, f (u1)=2Tn;
f(ui) = 2(T1+T2+· · · +Ti−1)+2Tn+i−1+ 1 for 2 ≤i≤n.
112 P. Jeyanthi, M. Selvi and D. Ramya
For the vertex label f, the induced edge label f∗is as follows:
f∗(ei) = Tifor 1 ≤i≤n−1, f∗(e0
j) = Tn+j−1for 1 ≤j≤n.
Then the edge labels are the triangular numbers: T1, T2,· · · , T2n−1. Hence PnK1is a
triangular difference mean graph.
The triangular difference mean labeling of P5K1is shown in Figure 3.
rr
rr
r
r r r r
r
1
30
41
131
21
93
9
65
3
45
Figure 3
Theorem 2.4The bistar Bm,n(m≥1, n ≥1) is a triangular difference mean graph.
Proof Let V(Bm,n ) = {u, v, ui, vj: 1 ≤i≤m, 1≤j≤n}and E(Bm,n ) = {uv, uui, v vj:
1≤i≤m, 1≤j≤n}. Define f:V(Bm,n)−→ Z+as follows:
f(u)=1, f (v) = 3, f(ui)=2Ti+1 + 1 for 1 ≤i≤m;
f(vj) = 2Tm+j+1 + 3 for 1 ≤j≤n.
For the vertex label f, the induced edge label f∗is as follows:
f∗(uv) = T1, f ∗(uui) = Ti+1 for 1 ≤i≤m;
f∗(vvj) = Tm+j+1 for 1 ≤j≤n.
The induced edge labels are the first m+n+ 1 triangular numbers and hence Bm,n is a
triangular difference mean graph.
The triangular difference mean labeling of B4,5is shown in Figure 4.
r
r
r
r
r
r
r
r
r
rr
7
13
21
31
13
113
93
75
59
45
Figure 4
bbbbb
b
Theorem 2.5A graph obtained by joining the roots of different stars to a new vertex, is a
triangular difference mean graph.
Triangular Difference Mean Graphs 113
Proof Let K1,n1, K1,n2,· · · , K1,nkbe kstars. Let Gbe a graph obtained by joining the
central vertices of the stars to a new vertex u.
Assign 1 to u; 2T1+1,2T2+1,· · · ,2Tk+ 1 to the central vertices of the stars; 2Tk+1 +2T1+
1,2Tk+2 + 2T1+ 1,· · · ,2Tk+n1+ 2T1+ 1 to the pendant vertices of the first star; 2Tk+n1+1 +
2T2+ 1,2Tk+n1+2 + 2T2+ 1,· · · ,2Tk+n1+n2+ 2T2+ 1 to the pendant vertices of the second star
and so on, finally assign the numbers 2Tk+n1+n2+···+nk−1+1 + 2Tk+ 1,2Tk+n1+n2+···+nk−1+2 +
2Tk+ 1,· · · ,2Tk+n1+n2+··· ,+nk−1+nk+ 2Tk+ 1 to the pendant vertices of the last star. Then,
the edge labels are the triangular numbers T1, T2,· · · , Tk+n1+n2+···+nk−1+nkand also the vertex
labels are all different.
The triangular difference mean labeling of the tree given in Theorem 2.5 with
k= 3, n1= 4, n2= 5 and n3= 4 is shown in Figure 5.
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
23
33
45
59
3
1
13
7
163
285
139
253
117
223
97
195
79
Figure 5
Theorem 2.6A tree T(n, m), obtained by identifying a central vertex of a star with a pendant
vertex of a path , is a triangular difference mean graph.
Proof Let v0, v1, v2,· · · , vnbe the vertices of the path Pnhaving path length n(n≥1) and
u, u1, u2,· · · , umbe the vertices of the star K1,m. Let T(n, m) be a tree obtained by identifying
v0with u.
Define f:V(T(n, m)) −→ Z+as follows:
f(v0) = 1, f(ui)=2Ti+ 1 for 1 ≤i≤m,
f(vj) = 2(Tm+1 +Tm+2 +. . . +Tm+j) + 1 for 1 ≤j≤n.
For a vertex label f, the induced edge label f∗is as follows:
f∗(v0ui) = Tifor 1 ≤i≤m;
f∗(vj−1vj) = Tm+jfor 1 ≤j≤n.
Then the induced edge labels are the first m+ntriangular numbers. Hence the tree
T(n, m) admits a triangular difference mean labeling.
The triangular difference mean labeling of a tree T(3,7) is shown in Figure 6.
114 P. Jeyanthi, M. Selvi and D. Ramya
r r
r r
rr
r
r
r
r
r
Figure 6
1
73
163
273
57
43
31
21
13
7
3
Theorem 2.7The caterpillar S(n, n, · · · , n
| {z }
mtimes
)is a triangular difference mean graph.
Proof Let v1, v2,· · · , vmbe the vertices of the path Pmand vi
j(1 ≤i≤n, 1≤j≤m) be
the pendant vertices incident with vj(1 ≤j≤m).
Then V(S(n, n, · · · , n
| {z }
m times
)) = {vj, vj
i: 1 ≤i≤n, 1≤j≤m}and E(S(n, n, · · · , n
| {z }
m times
)) =
{vtvt+1, vjvi
j: 1 ≤t≤m−1,1≤i≤n, 1≤j≤m}.
Define f:V(S(n, n, · · · , n
| {z }
m times
)) −→ Z+as follows:
f(v1)=1, f(vj) = 2(T1+T2+· · · +Tj−1) + 1 for 2 ≤j≤m;
f(vi
j) = f(vj)+2Tm+(j−1)n+i−1for 1 ≤j≤mand 1 ≤i≤n.
For each vertex label f, the induced edge label f∗is as follows:
f∗(vjvj+1) = Tjfor 1 ≤j≤m−1;
f∗(vjvi
j) = Tm+(j−1)n+i−1for 1 ≤j≤mand 1 ≤i≤n.
Then the edge labels are the triangular numbers T1, T2,· · · , Tm−1, Tm,· · · , Tm+n−1and
also the vertex labels are different. Hence S((n, n, · · · , n
| {z }
m times
)) is a triangular difference mean
graph.
Theorem 2.8Every cycle Cn(n > 3) is a triangular difference mean graph.
Triangular Difference Mean Graphs 115
Proof We prove this theorem in two cases.
Case 1. n = 4m+ 1.
Let S=1
2
n
P
i=1
Ti. Select some of the T0s
inamely Tl1, Tl2,· · · , Tlkfrom T1, T2,· · · , Tn
such that
k
P
i=1
Tli=S, where k < n and assume Tl1> Tl2>· · · > Tlk. Then the remaining
T0s
inamely, Tlk+1 , Tlk+2 ,· · · , Tlnare such that Tlk+1 > Tlk+2 >· · · , > Tlnand
n
P
i=k+1
Tli=
S−1. Let v1, v2,· · · , vk−1, vk, vk+1,· · · , vnbe the vertices of Cn. Label the first k+ 1 vertices
v1, v2, . . . , vk+1 as follows:
f(v1)=1, f(v2) = 2Tl1, f (v3)=2Tl1+ 2Tl2−1;
f(v4)=2Tl1+ 2Tl2+ 2Tl3−1,· · · , f (vk+1)=2Tl1+ 2Tl2+· · · + 2Tlk−1 and then,
f(vk+2) = 2Tl1+ 2Tl2+· · · + 2Tlk−2Tlk+1 −1;
f(vk+3) = 2Tl1+ 2Tl2+· · · + 2Tlk−2Tlk+1 −2Tlk+2 −1,· · · ;
f(vn) = 2Tl1+ 2Tl2+· · · + 2Tlk−2Tlk+1 −2Tlk+2 − · · · − 2Tln−1−1.
Hence, the edge labels are the triangular numbers {Tl1, Tl2,· · · , Tlk−1, Tlk, Tlk+1 ,· · · , Tln}=
{T1, T2,· · · , Tn}and also the vertex labels are all different.
Case 2. n 6= 4m+ 1, m >1.
Let S=1
2
n
P
i=1
Ti. Select some of the T0s
inamely Tl1, Tl2,· · · , Tlkfrom T1, T2,· · · , Tn
such that
k
P
i=1
Tli=S, where k < n and assume Tl1> Tl2>· · · > Tlk. Then the remaining T0s
i
namely, Tlk+1 , Tlk+2 ,· · · , Tlnare such that Tlk+1 > Tlk+2 >· · · > Tlnand
n
P
i=k+1
Tli=S. Let
v1, v2,· · · , vk−1, vk, vk+1 ,· · · , vnbe the vertices of Cn. We label the vertices v1, v2,· · · , vnas
follows:
f(v1)=1, f(v2) = 2Tl1+ 1, f (v3) = 2Tl1+ 2Tl2+ 1;
f(v4)=2Tl1+ 2Tl2+ 2Tl3+ 1,· · · ;
f(vk+1)=2Tl1+ 2Tl2+· · · + 2Tlk+ 1;
f(vk+2)=2Tl1+ 2Tl2+· · · + 2Tlk−2Tlk+1 + 1;
f(vk+3)=2Tl1+ 2Tl2+· · · , Tlk−2Tlk+1 −2Tlk+2 + 1,· · · ;
f(vn)=2Tl1+ 2Tl2+· · · + 2Tlk−2Tlk+1 −2Tlk+2 − · · · − 2Tln−1+ 1.
Thus, the edge labels are the triangular numbers {Tl1, Tl2,· · · , Tlk−1, Tlk, Tlk+1 ,· · · , Tln}
and also the vertex labels are all different.
The triangular difference mean labeling of C6is shown in Figure 8.
116 P. Jeyanthi, M. Selvi and D. Ramya
s
s
s
s
s
s
1
43
55
57
27
7
Figure 8
Theorem 2.9The graph Cn@Pm(n≥4, m ≥2) is a triangular difference mean graph.
Proof Let v1, v2,· · · , vnbe the vertices of the cycle Cnand u1, u2,· · · , umbe the vertices of
the path Pm. The graph Cn@Pmis obtained by identifying the vertexu1with the vertex v1. We
label the vertices of Cnas in Theorem 2.9 and assign the number 2Tn+1+2Tn+2+· · ·+2Tn+j−1+1
to vertex ujof the path Pmfor 2 ≤j≤m. Then the induced edge labels are the first m+n−1
triangular numbers. Hence, Cn@Pmis a triangular difference mean graph.
The triangular difference mean labeling of C4@P3is shown in Figure 9.
rr
r
r
r
r
3
131 73
21
9
Figure 9
Theorem 2.10 The cycle C3is not a triangular difference mean graph.
Proof Suppose C3is a triangular difference mean graph with triangular difference mean
labeling f. Let the vertices of C3be u, v , w. Let f(u) = x. Then to get 1 as an edge label we
must have f(v)∈ {x+ 1, x + 2, x −1, x −2}. To get T2, f (w)∈ {x+ 5, x + 6, x −5, x −6}or
f(w)∈ {x−6, x −7, x + 4, x + 5}. Then we get either {1,3,2}or {1,3,4}as the set of induced
edge labels. Therefore, T3= 6 can not be an edge label of C3. Hence C3is not a triangular
difference mean graph.
Triangular Difference Mean Graphs 117
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