ADCSS-Labeling of Shadow Graph of Some Graphs
Mathew Varkey T. K.
1and Sunoj B. S.
21
Department of Mathematics,
T K M College of Engineering, Kollam 5, INDIA.
2
Department of Mathematics,
Government Polytechnic College, Attingal, INDIA.
email:mathewvarkeytk@gmail.com, spalazhi@yahoo.com (Received on: November 12, 2016)
ABSTRACT
A graph labeling is an assignment of integers to the vertices or edges or both subject to certain conditions. In this paper, we introduce the new concept, an absolute difference of cubic and square sum labeling of a graph. The graph for which every edge label is the absolute difference of the sum of the cubes of the end vertices and the sum of the squares of the end vertices. It is also observed that the weights of the edges are found to be multiples of 2. Here we characterize shadow graphs of paths, cycles, stars, bistars, coconut tree, comb and triangular snakes for adcss labeling.
Keywords: Graph labeling, sum square graph, square sum graphs, cubic graph, shadow graph.
INTRODUCTION
All graphs in this paper are finite and undirected. The symbol V(G) and E(G) denotes the vertex set and edge set of a graph G. The graph whose cardinality of the vertex set is called the order of G, denoted by p and the cardinality of the edge set is called the size of the graph G, denoted by q. A graph with p vertices and q edges is called a (p,q)- graph.
A graph labeling is an assignment of integers to the vertices or edges. Some basic notations and definitions are taken from
1,2 and 3. Some basic concepts are taken from Frank Harary
1. We introduced the new concept, an absolute difference of cubic and square sum labeling of a graph
4. In
4,5,6,7,8,9,10,11, it is shown that planar grid, web graph, kayak paddle graph, snake graphs, friendship graph, armed crown, fan graph, cycle graphs ,wheel graph ,2-tuple graphs of some graphs, total graph of some graphs etc have an adcss labeling. In this paper we investigated ADCSS labeling of shadow graph of some graphs.
Definition: 1.1 [6]Let G = (V(G), E(G)) be a graph. A graph G is said to be absolute difference
of the sum of the cubes of the vertices and the sum of the squares of the vertices, if there exist
a bijection
f : V(G) → {1,2,---,p} such that the induced function 𝑓
𝑎𝑑𝑐𝑠𝑠∗: E(G) → multiples of 2 is given by 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑢𝑣) = (f(u)
3+f(v)
3) - ( f(u)
2+f(v)
2) is injective.
Definition: 1.2 A graph in which every edge associates distinct values with multiples of 2 is called the sum of the cubes of the vertices and the sum of the squares of the vertices. Such a labeling is called an absolute difference of cubic and square sum labeling or an absolute difference css-labeling.
MAIN RESULTS
Definition 2.1 The shadow graph D
2(G) of a connected graph G is constructed by taking two copies of G say G
1and G
2join each vertex v in G
1to the neighbors of the corresponding vertex u in G
2.
Theorem: 2.1 Shadow graph of path P
nadmits ADCSS - labeling.
Proof : Let G = D
2{P
n} and let v
1,v
2,---,v
2nare the vertices of G.
Here V(G) = 2n and E(G) = 4n-4
Define a function f : V → {1,2,3,---,2n} by f(v
i) = i , i = 1,2,---,2n.
For the vertex labeling f, the induced edge labeling 𝑓
𝑎𝑑𝑐𝑠𝑠∗is defined as follows
𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑖𝑣
𝑖+1) = (i+1)
2i+i
2(i-1), i = 1,2,3,---,n-1 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑛+𝑖𝑣
𝑛+𝑖+1) = (n+i)
2(n+i-1) + (n+i+1)
2(n+i) , i = 1,2,3,---,n-1 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑖𝑣
𝑛+𝑖+1) = (i)
2(i-1) + (n+i+1)
2(n+i) , i = 1,2,3,---,n-1 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑖+1𝑣
𝑛+𝑖) = (i+1)
2(i) + (n+i)
2(n+i-1) , i = 1,2,3,---,n-1
All edge values of G are distinct, which are multiples of 2.That is the edge values of G are in the form of an increasing order. Hence D
2{P
n} admits adcss-labeling.
Example 2.1 G = D
2{P
5}
Fig.i
Theorem: 2.2 Shadow graph of cycle C
nadmits ADCSS - labeling.
Proof : Let G = D
2{C
n} and let v
1,v
2,---,v
2nare the vertices of G.
Here |V(G)| = 2n and |E(G) | = 4n
Define a function f : V → {1,2,3,---,2n} by f(v
i) = i , i = 1,2,---,2n.
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For the vertex labeling f, the induced edge labeling 𝑓
𝑎𝑑𝑐𝑠𝑠∗is defined as follows
𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑖𝑣
𝑖+1) = (i+1)
2i+i
2(i-1), i = 1,2,3,---,n-1 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑛+𝑖𝑣
𝑛+𝑖+1) = (n+i)
2(n+i-1) + (n+i+1)
2(n+i), i = 1,2,3,---,n-1 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑖𝑣
𝑛+𝑖+1) = (i)
2(i-1) + (n+i+1)
2(n+i) , i = 1,2,3,---,n-1 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑖+1𝑣
𝑛+𝑖) = (i+1)
2(i) + (n+i)
2(n+i-1) , i = 1,2,3,---,n-1 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
1𝑣
𝑛) = n
2(n-1)
𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
1𝑣
2𝑛) = (2n)
2(2n-1) 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑛𝑣
𝑛+1) = n
2(n-1) + (n+1)
2n 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑛𝑣
2𝑛) = n
2(n-1) + (2n)
2(2n-1)
All edge values of G are distinct, which are multiples of 2.That is the edge values of G are in the form of an increasing order. Hence D
2{C
n} admits adcss-labeling.
Example : 2.2 Let G = D
2{C
5}
Fig.ii
Theorem: 2.3 Shadow graph of comb graph admits ADCSS - labeling.
Proof : Let G = D
2{Comb(n)} and let v
1,v
2,---,v
4nare the vertices of G.
Here |V(G)| = 4n and |E(G)| = 8n - 4 Define a function f : V → {1,2,3,---,4n} by
f(v
i) = i , i = 1,2,---,4n.
For the vertex labeling f, the induced edge labeling 𝑓
𝑎𝑑𝑐𝑠𝑠∗is defined as follows
𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑖𝑣
𝑖+1) = (i+1)
2i+i
2(i-1), i = 1,2,3,---,n+1 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑛+2+𝑖𝑣
𝑛+3+𝑖) = (n+2+i)
2(n+1+i) + (n+3+i)
2(n+2+i) , i = 1,2,3,---,n+1 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑖𝑣
𝑛+3+𝑖) = (i)
2(i-1) + (n+3+i)
2(n+2+i) , i = 1,2,3,---,n+1 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑖+1𝑣
𝑛+2+𝑖) = (i+1)
2(i) + (n+2+i)
2(n+1+i) , i = 1,2,3,---,n+1 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑖+2𝑣
2𝑛+3+2𝑖) = (i+2)
2(i+1) + (2n+3+2i)
2(2n+2+2i) , i = 1,2,3,---,n-2 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑖+2𝑣
2𝑛+4+2𝑖) = (i+2)
2(i+1) + (2n+4+2i)
2(2n+3+2i) , i = 1,2,3,---,n-2 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑛+4+𝑖𝑣
2𝑛+3+2𝑖) = (n+4+i)
2(n+3+i) + (2n+3+2i)
2(2n+2+2i), i = 1,2,3,---,n-2 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑛+4+𝑖𝑣
2𝑛+4+2𝑖) = (n+4+i)
2(n+3+i) + (2n+4+2i)
2(2n+3+2i) , i = 1,2,3,---,n-2 All edge values of G are distinct, which are multiples of 2.That is the edge values of G are in the form of an increasing order. Hence D
2{Comb(n)} admits adcss-labeling.
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fig – ii
Theorem: 2.4 Shadow graph of star graph admits ADCSS - labeling.
Proof : Let G = D
2{K
1,n} and let v
1,v
2,---,v
2n+2are the vertices of G.
Here |V(G)| = 2n+2 and |E(G)| = 4n
Define a function f : V → {1,2,3,---,2n+2} by f(v
i) = i , i = 1,2,---,2n+2.
For the vertex labeling f, the induced edge labeling 𝑓
𝑎𝑑𝑐𝑠𝑠∗is defined as follows
𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑖𝑣
2𝑛+2) = (2n+2)
2(2n+1)+i
2(i-1), i = 1,2,3,---,2n 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑖𝑣
2𝑛+1) = (2n+1)
2(2n)+i
2(i-1), i = 1,2,3,---,2n All edge values of G are distinct, which are multiples of 2.That is the edge values of G are in the form of an increasing order. Hence D
2{K
1,n} admits adcss-labeling.
Theorem: 2.5 Shadow graph of coconut tree graph admits ADCSS - labeling.
Proof : Let G = D
2{CT(m,n)} and let v
1,v
2,---,v
2n+2mare the vertices of G.
Here |V(G) | = 2n+2m and | E(G) | = 4(m+n-1) Define a function f : V → {1,2,3,---,2n+2m} by
f(v
i) = i , i = 1,2,---,2n+2m.
For the vertex labeling f, the induced edge labeling 𝑓
𝑎𝑑𝑐𝑠𝑠∗is defined as follows
𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑖𝑣
𝑖+1) = (i+1)
2(i)+i
2(i-1), i = 1,2,3,--- ---,m-1
𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑚+𝑖𝑣
𝑚+𝑖+1) = (m+i)
2(m+i-1)+(m+i+1)
2(m+i), i = 1,2,3,---,m-1 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑖𝑣
𝑚+𝑖+1) = (i)
2(i-1)+(m+i+1)
2(m+i), i = 1,2,3,---,m-1 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑖+1𝑣
𝑚+𝑖) = (i+1)
2(i)+(m+i)
2(m+i-1), i = 1,2,3,---,m-1 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
2𝑚+𝑖𝑣
𝑚+1) = (2m+i)
2(2m+i-1)+(m+1)
2(m), i = 1,2,3,---,2n 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
2𝑚+𝑖𝑣
1) = (2m+i)
2(2m+i-1), i = 1,2,3,---,2n All edge values of G are distinct, which are multiples of 2.That is the edge values of G are in the form of an increasing order. Hence D
2{CT(m,n)} admits adcss-labeling.
Theorem: 2.6 Shadow graph of bistar graph admits ADCSS - labeling.
Proof : Let G = D
2{B(m,n)} and let v
1,v
2,---,v
2n+2mare the vertices of G.
Here |V(G)| = 2n+2m+4 and |E(G) | = 4m+4n+2 Define a function f : V → {1,2,3,---,2n+2m} by
f(v
i) = i , i = 1,2,---,2n+2m.
For the vertex labeling f, the induced edge labeling 𝑓
𝑎𝑑𝑐𝑠𝑠∗is defined as follows
𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑖𝑣
2𝑚+1) = (2m+1)
2(2m)+i
2(i-1), i = 1,2,3,---,2m 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑖𝑣
2𝑚+2) = (2m+2)
2(2m+1)+i
2(i-1), i = 1,2,3,---,2m
𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
2𝑚+2+𝑖𝑣
2𝑚+2𝑛+3) = (2m+2+i)
2(2m+1+i)+(2m+2n+3)
2(2m+2n+2),i = 1,2,3,---,2n 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
2𝑚+2+𝑖𝑣
2𝑚+2𝑛+4) = (2m+2+i)
2(2m+1+i)+(2m+2n+4)
2(2m+2n+3),i = 1,2,3,---,2n 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
2𝑚+1𝑣
2𝑚+2𝑛+3) = (2m+1)
2(2m)+(2m+2n+3)
2(2m+2n+2)
𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
2𝑚+2𝑣
2𝑚+2𝑛+4) = (2m+2)
2(2m+1)+(2m+2n+4)
2(2m+2n+3)
All edge values of G are distinct, which are multiples of 2.That is the edge values of
G are in the form of an increasing order. Hence D
2{B(m,n)} admits adcss-labeling.
Theorem: 2.7 Shadow graph of triangular snake graphs admits ADCSS - labeling.
Proof : Let G = D
2{T
n} and let v
1,v
2,---,v
4n-2are the vertices of G.
Here |V(G) | = 4n-2 and |E(G) | = 12n-12 Define a function f : V → {1,2,3,---,4n-2} by
f(v
i) = i , i = 1,2,---,4n-2.
For the vertex labeling f, the induced edge labeling 𝑓
𝑎𝑑𝑐𝑠𝑠∗is defined as follows
𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑖𝑣
2𝑛+𝑖) = (2n+i)
2(2n+i-1)+i
2(i-1), i = 1,2,3,---,n-1 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑖+1𝑣
2𝑛+𝑖) = (2n+i)
2(2n+i-1)+(i+1)
2(i), i = 1,2,3,---,n-1 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑛+𝑖𝑣
3𝑛−1+𝑖) = (3n-1+i)
2(3n-2+i)+(n+i)
2(n+i-1), i = 1,2,3,---,n-1 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑛+𝑖+1𝑣
3𝑛−1+𝑖) = (3n-1+i)
2(3n-2+i)+(n+i+1)
2(n+i), i = 1,2,3,---,n-1 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑖𝑣
3𝑛−1+𝑖) = (3n-1+i)
2(3n-2+i)+(i)
2(i-1), i = 1,2,3,---,n-1 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑖+1𝑣
3𝑛−1+𝑖) = (3n-1+i)
2(3n-2+i)+(i+1)
2(i), i = 1,2,3,---,n-1 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑛+𝑖𝑣
2𝑛+𝑖) = (2n+i)
2(2n+i-1)+(n+i)
2(n+i-1), i = 1,2,3,---,n-1 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑛+𝑖+1𝑣
2𝑛+𝑖) = (2n+i)
2(2n+i-1)+(n+i+1)
2(n+i), i = 1,2,3,---,n-1 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑖𝑣
𝑖+1) = (i+1)
2(i)+i
2(i-1), i = 1,2,3,---,n-1 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑛+𝑖𝑣
𝑛+𝑖+1) = (n+i)
2(n+i-1)+(n+i+1)
2(n+i), i = 1,2,3,---,n-1 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑖𝑣
𝑛+𝑖+1) = (i)
2(i-1)+(n+i+1)
2(n+i), i = 1,2,3,---,n-1 𝑓
𝑎𝑑𝑐𝑠𝑠∗(𝑣
𝑖+1𝑣
𝑛+𝑖) = (i+1)
2(i)+(n+i)
2(n+i-1), i = 1,2,3,---,n-1
All edge values of G are distinct, which are multiples of 2.That is the edge values of G are in the form of an increasing order. Hence D
2{T
n} admits adcss-labeling.
Example : 2.2 Let G = D
2{T
4}
Fig.iii
REFERENCES
1. F Harary, Graph Theory, Addison-Wesley, Reading, Mass, (1972).
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