Let G=(V,E) be a graph with p vertices and q edges. Let f:V?{1,2,"¦q+1} is called an Integral Root labeling if it is possible to label all the vertices v?V with distinct elements from {1,2,"¦q+1} such that it induces an edge labeling f^+:E?{1,2,"¦q} defined as f^+ (uv)=?v((?(f(u))?^2+?(f(v))?^2+f(u)f(v))3)? is distinct for all uv?E. (i.e.) The distinct vertex labeling induces a distinct edge labeling on the graph. The graph which admits Integral Root labeling is called an Integral Root Graph. In this paper, we investigate the Integral Root labeling of P_m?G graphs likeP_m?P_n,P_m?(P_n ?K_1), P_m?L_n,P_m?(P_n ?K_1,2), P_m?(P_n ?K_1,3), P_m?(P_n ?K_1)?K_1,2 V. L. Stella Arputha Mary | N. Nanthini"Integral Root Labeling of Pm ∪ G Graphs" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-5 , August 2018, URL: http://www.ijtsrd.com/papers/ijtsrd18233.pdf http://www.ijtsrd.com/mathemetics/other/18233/integral-root-labeling-of-pm-∪-g-graphs/v-l-stella-arputha-mary

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ISSN No: 2456
International
Research
Integral Root Labeling o
V. L. Stella Arputha Mary
1
Department of Mathematics,
St. Mary’s College (Autonomous), Thoothukudi
ABSTRACT
Let ‫ܩ‬ ൌ ሺܸ, ‫ܧ‬ሻ be a graph with ‫݌‬ vertices and
labeling if it is possible to label all the vertices
induces an edge labeling ݂ା
: ‫ܧ‬ → ሼ1,2, …
݂ା
ሺ‫ݒݑ‬ሻ ൌ ቜට
ሺ௙ሺ௨ሻሻమାሺ௙ሺ௩ሻሻమା௙ሺ௨ሻ௙ሺ௩ሻ
ଷ
ቝ is distinct for all
distinct edge labeling on the graph. The graph which
Graph.
In this paper, we investigate the Integral Root labeling of
‫ܮ‬௡,ܲ௠ ∪ ሺܲ௡ʘ‫ܭ‬ଵ,ଶሻ, ܲ௠ ∪ ሺܲ௡ʘ‫ܭ‬ଵ,ଷሻ, ܲ௠
Key words: P_m∪P_n, P_m∪(P_n ʘK_1), P_m
ʘK_1), P_m∪(P_n ʘK_1)ʘK_1,2
INTRODUCTION
The graph considered here will be finite, undirected and simple. The vertex set is denoted by
edge set is denoted by‫ܧ‬ሺ‫ܩ‬ሻ. For all detailed survey of gr
terminology and notations we follow Haray [
concept of Integral Root Labeling of graphs in [8]. In this paper we investigate Integral Root la
ܲ௠ ∪ ‫ ܩ‬graphs. The definitions and other informations which are useful for the present investigation are given
below.
BASIC DEFINITIONS
Definition: 3.1
A walk in which ‫ݑ‬ଵ, ‫ݑ‬ଶ, … ‫ݑ‬௡ are distinct is called a
Definition: 3.2
The graph obtained by joining a single pendent edge to each vertex of a path is called a
Definition: 3.3
The Cartesian product of two graphs ‫ܩ‬
vertices ‫ݑ(=ݑ‬1‫ݑ‬2) and ‫ݒ(=ݒ‬1‫ݒ‬2) are adjacent in
is adjacent to ‫ݒ‬1) .It is denoted by ‫ܩ‬1×‫ܩ‬2
Definition: 3.4
The Corona of two graphs ‫ܩ‬1 and ‫ܩ‬2 is the graph
of ‫ܩ‬2 where the ith
vertex of ‫ܩ‬1 is adjacent to every vertex in the
@ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 5 | Jul-Aug 2018
ISSN No: 2456 - 6470 | www.ijtsrd.com | Volume
International Journal of Trend in Scientific
Research and Development (IJTSRD)
International Open Access Journal
Integral Root Labeling of Pm∪∪∪∪G Graphs
V. L. Stella Arputha Mary1
, N. Nanthini2
Department of Mathematics, 2
M.phil Scholar
St. Mary’s College (Autonomous), Thoothukudi, Tamil Nadu, India
vertices and ‫ ݍ‬edges. Let ݂: ܸ → ሼ1,2, … ‫ݍ‬ ൅ 1ሽ is called an
if it is possible to label all the vertices ‫ݒ‬ ∈ ܸ with distinct elements from ሼ1
ሼ … ‫ݍ‬ሽ defined as
ቝ is distinct for all ‫ݒݑ‬ ∈ ‫( .ܧ‬i.e.) The distinct vertex labeling induces a
distinct edge labeling on the graph. The graph which admits Integral Root labeling is called an
stigate the Integral Root labeling of ܲ௠ ∪ ‫ ܩ‬graphs likeܲ௠ ∪
௠ ∪ ሺܲ௡ʘ‫ܭ‬ଵሻʘ‫ܭ‬ଵ,ଶ
K_1), P_m∪L_n, P_m∪(P_n ʘK_1,2), P_m∪(P_n
The graph considered here will be finite, undirected and simple. The vertex set is denoted by
. For all detailed survey of graph labeling we refer to Gallian [1]. For all standard
we follow Haray [2]. V.L Stella Arputha Mary and N.Nanthini introduced the
concept of Integral Root Labeling of graphs in [8]. In this paper we investigate Integral Root la
graphs. The definitions and other informations which are useful for the present investigation are given
are distinct is called a Path. A path on ݊ vertices is denoted by
The graph obtained by joining a single pendent edge to each vertex of a path is called a
‫ܩ‬1=(ܸ1,‫ܧ‬1) and ‫ܩ‬2=(ܸ2,‫ܧ‬2) is a graph ‫)ܧ,ܸ(=ܩ‬ with
) are adjacent in ‫ܩ‬1×‫ܩ‬2 whenever (‫ݑ‬1=‫ݒ‬1and ‫ݑ‬2 is adjacent to
2.
is the graph ‫ܩ=ܩ‬1⨀‫ܩ‬2 formed by taking one copy of
is adjacent to every vertex in the ith
copy of ‫ܩ‬2.
Aug 2018 Page: 2141
com | Volume - 2 | Issue – 5
Scientific
(IJTSRD)
International Open Access Journal
G Graphs
India
ሽ is called an Integral Root
ሼ1,2, … ‫ݍ‬ ൅ 1ሽ such that it
(i.e.) The distinct vertex labeling induces a
admits Integral Root labeling is called an Integral Root
ܲ௡,ܲ௠ ∪ ሺܲ௡ʘ‫ܭ‬ଵሻ, ܲ௠ ∪
(P_n ʘK_1,3), P_m∪(T_n
The graph considered here will be finite, undirected and simple. The vertex set is denoted by ܸሺ‫ܩ‬ሻ and the
aph labeling we refer to Gallian [1]. For all standard
2]. V.L Stella Arputha Mary and N.Nanthini introduced the
concept of Integral Root Labeling of graphs in [8]. In this paper we investigate Integral Root labeling of
graphs. The definitions and other informations which are useful for the present investigation are given
oted by ܲ௡
The graph obtained by joining a single pendent edge to each vertex of a path is called a Comb.
) with ܸ=ܸ1×ܸ2 and two
is adjacent to ‫ݒ‬2) or (‫ݑ‬2=‫ݒ‬2and ‫ݑ‬1
rmed by taking one copy of ‫ܩ‬1 and |(‫ܩ‬1)| copies

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Definition: 3.5
The product graph ܲଶ × ܲ௡ is called a Ladder and it is denoted by ‫ܮ‬௡
Definition: 3.6
The union of two graphs ‫ܩ‬ଵ ൌ ሺܸଵ, ‫ܧ‬ଵ) and ‫ܩ‬ଶ = (ܸଶ, ‫ܧ‬ଶ) is a graph ‫ܩ‬ = ‫ܩ‬ଵ ∪ ‫ܩ‬ଶ with vertex set ܸ = ܸଵ ∪ ܸଶ
and the edge set ‫ܧ‬ = ‫ܧ‬ଵ ∪ ‫ܧ‬ଶ.
Definition: 3.7
The graph ܲ௡ʘ‫ܭ‬ଵ,ଶ is obtained by attaching ‫ܭ‬ଵ,ଶ to each vertex of ܲ௡.
Definition: 3.8
The graph ܲ௡ʘ‫ܭ‬ଵ,ଷ is obtained by attaching ‫ܭ‬ଵ,ଷ to each vertex of ܲ௡.
Definition: 3.9
A graph that is not connected is disconnected. A graph ‫ܩ‬ is said to be disconnected if there exist two nodes in
‫ܩ‬ such that no path in ‫ܩ‬ has those nodes as endpoints. A graph with just one vertex is connected. An edgeless
graph with two (or) more vertices is disconnected
MAIN RESULTS
Theorem: 4.1
ܲ௠ ∪ ܲ௡ is an Integral Root graph.
Proof:
Let ܲ௠ = ‫ݑ‬ଵ, ‫ݑ‬ଶ, … . , ‫ݑ‬௠ be a path on ݉ vertices.
Let ܲ௡ = ‫ݒ‬ଵ, ‫ݒ‬ଶ, … . , ‫ݒ‬௡ be another one path on ݊ vertices.
Let ‫ܩ‬ = ܲ௠ ∪ ܲ௡.
Define a function ݂: ܸ(‫)ܩ‬ → {1,2, … , ‫ݍ‬ + 1} by
݂(‫ݑ‬௜) = ݅; 1 ≤ ݅ ≤ ݉;
݂(‫ݒ‬௜) = ݉ + ݅; 1 ≤ ݅ ≤ ݊.
Then we find the edge labels
݂ା(‫ݑ‬௜‫ݑ‬௜ାଵ) = ݅; 1 ≤ ݅ ≤ ݉ − 1;
݂ା(‫ݒ‬௜‫ݒ‬௜ାଵ) = ݉ + ݅; 1 ≤ ݅ ≤ ݊ − 1.
Then the edge labels are distinct.
Hence ܲ௠ ∪ ܲ௡ is a Integral Root graph.
Example: 4.2
An Integral Root labeling of ܲହ ∪ ܲ଺ is show below.
Figure: 1
Theorem: 4.3
ܲ௠ ∪ (ܲ௡ʘ‫ܭ‬ଵ) is a Integral Root graph.

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Proof:
Let ܲ௡ʘ‫ܭ‬ଵ be a Comb graph obtained from a path ܲ௡ = ‫ݒ‬ଵ, ‫ݒ‬ଶ, … . , ‫ݒ‬௡ by joining a vertex ‫ݑ‬௜ to ‫ݒ‬௜, 1 ≤ ݅ ≤ ݊.
Let ܲ௠ = ‫ݓ‬ଵ, ‫ݓ‬ଶ, … , ‫ݓ‬௠ be a path.
Let ‫ܩ‬ = ܲ௠ ∪ (ܲ௡ʘ‫ܭ‬ଵ).
Define a function ݂: ܸ(‫)ܩ‬ → {1,2, … . , ‫ݍ‬ + 1} by
݂(‫ݓ‬௜) = ݅; 1 ≤ ݅ ≤ ݉;
݂(‫ݒ‬௜) = ݉ + 2݅ − 1; 1 ≤ ݅ ≤ ݊;
݂(‫ݑ‬௜) = ݉ + 2݅; 1 ≤ ݅ ≤ ݊.
Then we find the edge labels are
݂ା(‫ݓ‬௜‫ݓ‬௜ାଵ) = ݅; 1 ≤ ݅ ≤ ݉ − 1;
݂ା(‫ݒ‬௜‫ݒ‬௜ାଵ) = ݉ + 2݅; 1 ≤ ݅ ≤ ݊ − 1;
݂ା(‫ݒ‬௜‫ݑ‬௜) = ݉ + 2݅ − 1; 1 ≤ ݅ ≤ ݊ − 1.
Then the edge labels are distinct.
Hence ܲ௠ ∪ (ܲ௡ʘ‫ܭ‬ଵ) is an Integral Root graph.
Example: 4.4
An Integral Root labeling of ܲହ ∪ (ܲହʘ‫ܭ‬ଵ) is given below.
Figure: 2
Theorem: 4.5
ܲ௠ ∪ ‫ܮ‬௡ is an Integral Root graph.
Proof:
Let ܲ௠ = ‫ݑ‬ଵ, ‫ݑ‬ଶ, … . , ‫ݑ‬௠ be a path.
Let {‫ݒ‬ଵ, ‫ݒ‬ଶ, … , ‫ݒ‬௡ , ‫ݓ‬ଵ, ‫ݓ‬ଶ, … . . , ‫ݓ‬௡} be the vertices of ladder.
efine a function ݂: ܸ(‫)ܩ‬ → {1,2, … . , ‫ݍ‬ + 1} by
݂(‫ݑ‬௜) = ݅; 1 ≤ ݅ ≤ ݉;
݂(‫ݒ‬௜) = ݉ + 3݅ − 2; 1 ≤ ݅ ≤ ݊;
݂(‫ݓ‬௜) = ݉ + 3݅ − 1; 1 ≤ ݅ ≤ ݊.

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Then we find the edge labels
݂ା(‫ݑ‬௜‫ݑ‬௜ାଵ) = ݅; 1 ≤ ݅ ≤ ݉ − 1;
݂ା(‫ݒ‬௜‫ݒ‬௜ାଵ) = ݉ + 3݅ − 1; 1 ≤ ݅ ≤ ݊ − 1;
݂ା(‫ݓ‬௜‫ݓ‬௜ାଵ) = ݉ + 3݅; 1 ≤ ݅ ≤ ݊ − 1.
݂ା(‫ݒ‬௜‫ݓ‬௜) = ݉ + 3 − 2݅; 1 ≤ ݅ ≤ ݊.
Then the edge labels are distinct.
Hence ܲ௠ ∪ ‫ܮ‬௡ is an Integral Root graph.
Example: 4.6
An Integral Root labeling of ܲହ ∪ ‫ܮ‬ହ is displayed below.
Figure: 3
Theorem: 4.7
ܲ௠ ∪ (ܲ௡ʘ‫ܭ‬ଵ,ଶ) is a Integral Root graph.
Proof:
Let ܲ௡ʘ‫ܭ‬ଵ,ଶ be a graph obtained by attaching each vertex of a path ܲ௡ to the central vertex of ‫ܭ‬ଵ,ଶ.
Let ܲ௡ = ‫ݒ‬ଵ, ‫ݒ‬ଶ, … , ‫ݒ‬௡ be a path.
Let ‫ݓ‬௜ and ‫ݔ‬௜ be the vertices of ‫ܭ‬ଵ,ଶ which are attaching with the vertex ‫ݒ‬௜ of ܲ௡
1 ≤ ݅ ≤ ݊ .
Let ܲ௠ = ‫ݑ‬ଵ, ‫ݑ‬ଶ, … . , ‫ݑ‬௠ be a path.
Let ‫ܩ‬ = ܲ௠ ∪ (ܲ௡ʘ‫ܭ‬ଵ,ଶ).
Define a function ݂: ܸ(‫)ܩ‬ → {1,2, … . , ‫ݍ‬ + 1} by
݂(‫ݑ‬௜) = ݅; 1 ≤ ݅ ≤ ݉;
݂(‫ݒ‬௜) = ݉ + 3݅ − 1; 1 ≤ ݅ ≤ ݊;
݂(‫ݓ‬௜) = ݉ + 3݅ − 2; 1 ≤ ݅ ≤ ݊;
݂(‫ݔ‬௜) = ݉ + 3݅; 1 ≤ ݅ ≤ ݊.

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Then we find the edge labels are
݂ା(‫ݑ‬௜‫ݑ‬௜ାଵ) = ݅; 1 ≤ ݅ ≤ ݉ − 1;
݂ା(‫ݒ‬௜‫ݒ‬௜ାଵ) = ݉ + 3݅; 1 ≤ ݅ ≤ ݊ − 1;
݂ା(‫ݒ‬௜‫ݓ‬௜) = ݉ + 3݅ − 2; 1 ≤ ݅ ≤ ݊;
݂ା(‫ݔ‬௜‫ݒ‬௜) = ݉ + 3݅ − 1; 1 ≤ ݅ ≤ ݊.
Hence ܲ௠ ∪ (ܲ௡ʘ‫ܭ‬ଵ,ଶ) is a Integral Root graph.
Example: 4.8
An Integral Root labeling of ܲହ ∪ (ܲଷʘ‫ܭ‬ଵ,ଶ) is given below.
Figure: 4
Theorem: 4.9
ܲ௠ ∪ (ܲ௡ʘ‫ܭ‬ଵ,ଷ) is a Integral Root graph.
Proof:
Let ܲ௡ʘ‫ܭ‬ଵ,ଶ be a graph obtained by attaching each vertex of a path ܲ௡ to the central vertex of ‫ܭ‬ଵ,ଷ.
Let ܲ௡ = ‫ݓ‬ଵ, ‫ݓ‬ଶ, … , ‫ݓ‬௡ be a path.
Let ‫ݒ‬௜, ‫ݔ‬௜ and ‫ݕ‬௜ be the vertices of ‫ܭ‬ଵ,ଷ which are attaching with the vertex ‫ݓ‬௜ of ܲ௡
1 ≤ ݅ ≤ ݊ .
Let ܲ௠ = ‫ݑ‬ଵ, ‫ݑ‬ଶ, … . , ‫ݑ‬௠ be a path.
Let ‫ܩ‬ = ܲ௠ ∪ (ܲ௡ʘ‫ܭ‬ଵ,ଷ).
Define a function ݂: ܸ(‫)ܩ‬ → {1,2, … . , ‫ݍ‬ + 1} by
݂(‫ݑ‬௜) = ݅; 1 ≤ ݅ ≤ ݉;
݂(‫ݒ‬௜) = ݉ + 4݅ − 3; 1 ≤ ݅ ≤ ݊;
݂(‫ݓ‬௜) = ݉ + 4݅ − 2; 1 ≤ ݅ ≤ ݊;
݂(‫ݔ‬௜) = ݉ + 4݅ − 1; 1 ≤ ݅ ≤ ݊;
݂(‫ݕ‬௜) = ݉ + 4݅; 1 ≤ ݅ ≤ ݊.
Then we find the edge labels are
݂ା(‫ݑ‬௜‫ݑ‬௜ାଵ) = ݅; 1 ≤ ݅ ≤ ݉ − 1;
݂ା(‫ݓ‬௜‫ݓ‬௜ାଵ) = ݉ + 4݅; 1 ≤ ݅ ≤ ݊ − 1;
݂ା(‫ݒ‬௜‫ݓ‬௜) = ݉ + 4݅ − 3; 1 ≤ ݅ ≤ ݊;
݂ା(‫ݔ‬௜‫ݓ‬௜) = ݉ + 4݅ − 2; 1 ≤ ݅ ≤ ݊;
݂ା(‫ݕ‬௜‫ݓ‬௜) = ݉ + 4݅ − 1; 1 ≤ ݅ ≤ ݊.
Then the edge labels are distinct.
Hence ܲ௠ ∪ (ܲ௡ʘ‫ܭ‬ଵ,ଷ) is a Integral Root graph.

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Example: 4.10
An Integral Root labeling of ܲହ ∪ (ܲଷʘ‫ܭ‬ଵ,ଷ) is given below.
Figure: 5
Theorem: 4.11
ܲ௠ ∪ (ܲ௡ʘ‫ܭ‬ଵ)ʘ‫ܭ‬ଵ,ଶ is a Integral root graph.
Proof:
Let ‫ܩ‬ = ܲ௠ ∪ (ܲ௡ʘ‫ܭ‬ଵ)ʘ‫ܭ‬ଵ,ଶ.
Let ܲ௠ = ‫ݑ‬ଵ, ‫ݑ‬ଶ, … . , ‫ݑ‬௠ be a path.
Let ‫ܩ‬ଶ be a comb and ‫ܩ‬ଵ be the obtained by attaching ‫ܭ‬ଵ,ଶ at each pendant vertex of ‫ܩ‬ଶ.
Let its vertices be ‫ݒ‬௜, ‫ݓ‬௜, ‫ݔ‬௜, ‫ݕ‬௜ 1 ≤ ݅ ≤ ݊.
Define a function ݂: ܸ(‫)ܩ‬ → {1,2, … . , ‫ݍ‬ + 1} by
݂(‫ݑ‬௜) = ݅; 1 ≤ ݅ ≤ ݉;
݂(‫ݒ‬௜) = ݉ + 5݅ − 3; 1 ≤ ݅ ≤ ݊;
݂൫‫ݓ‬.
௜
൯ = ݉ + 5݅ − 4; 1 ≤ ݅ ≤ ݊;
݂(‫ݔ‬௜) = ݉ + 5݅ − 2; 1 ≤ ݅ ≤ ݊;
݂(‫ݕ‬௜) = ݉ + 5݅; 1 ≤ ݅ ≤ ݊.
Then we find edge labels are
݂ା(‫ݑ‬௜ାଵ‫ݑ‬௜) = ݅; 1 ≤ ݅ ≤ ݉ − 1;
݂ା(‫ݒ‬௜ାଵ‫ݒ‬௜) = ݉ + 5݅ − 1; 1 ≤ ݅ ≤ ݊ − 1;
݂ା(‫ݒ‬௜‫ݓ‬௜) = ݉ + 5݅ − 4; 1 ≤ ݅ ≤ ݊;
݂ା(‫ݓ‬௜‫ݔ‬௜) = ݉ + 5݅ − 3; 1 ≤ ݅ ≤ ݊;
݂ା(‫ݓ‬௜‫ݕ‬௜) = ݉ + 5݅ − 2; 1 ≤ ݅ ≤ ݊.
Then the edge labels are distinct.
Hence ܲ௠ ∪ (ܲ௡ʘ‫ܭ‬ଵ)ʘ‫ܭ‬ଵ,ଶ is an Integral Root graph.
Example: 4.12
An Integral Root labeling of ܲହ ∪ (ܲସʘ‫ܭ‬ଵ)ʘ‫ܭ‬ଵ,ଶ is given below.

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Figure: 6
REFERENCE
1. J. A. Gallian, 2010, “A dynamic Survey of graph labeling,” The electronic Journal of
Combinatories17#DS6.
2. F. Harary, 1988, “Graph Theory,” Narosa Publishing House Reading, New Delhi.
3. S. Sandhya, S. Somasundaram, S. Anusa, “Root Square Mean labeling of graphs,” International Journal of
Contemporary Mathematical Science, Vol.9, 2014, no.667-676.
4. S. S. Sandhya, E. Ebin Raja Merly and S. D. Deepa, “Heronian Mean Labeling of Graphs”, communicated
to International journal of Mathematical Form.
5. S. Sandhya, E. Ebin Raja Merly and S. D. Deepa, “ Some results On Heronian Mean Labeling of Graphs”,
communicated to Journal of Discrete Mathematical Science of crypotography.
6. S. Sandhya, S. Somasundaram, S. Anusa, “Root Square Mean labeling of Some Disconnected graphs,”
communicated to International Journal of Mathematical Combinatorics.
7. S. S. Sandhya, S.Somasundaram and A.S.Anusa, “Root Mean Labeling of Some New Disconnected
Graphs”, communicated to International journal of Mathematical Tends and Technology, Volume 15 no.2
(2014) Pg no. 85-92.
8. V. L Stella Arputha Mary, and N. Nanthini, “Integral Root labeling of graph” International Journal of
Mathematics Trends Technology(IJMTT), vol.54, no.6(2018), pp.437-442.