T Wiener Index of Fibonacci Labeled Graph P n ʘ F 4 R. Palanikumar * and A. Rameshkumar**
*Assistant Professor,
Department of Mathematics, Srimad Andavan Arts & Science College (Autonomous), No.7, Nelson Road, Trichy–620005, INDIA.
** Assistant Professor, Department of Mathematics,
Srimad Andavan Arts & Science College (Autonomous), No.7, Nelson Road, Trichy–620005, INDIA.
email: palanikumar2261982@gmail.com, andavanmathsramesh@gmail.com.
(Received on: November 1, 2018) ABSTRACT
A Graph is said to be Fibonacci labeling if it has induced edge labeling that satisfies certain conditions. The corona G
1ʘ G
2of two graphs G
1(with n
1vertices and m
1edges) and G
2(with n
2vertices and m
2edges) is defined as the graph obtained by taking one copy G
1and n
1copies of G
2and then joining the i
thvertex of G
1with an edge to every vertex in the i
thcopy of G
2. The Wiener index
1W is a graph index defined for a graph on n nodes by as the sum of the numbers of edges in the shortest paths in a graph between all pairs of vertices. A Wiener Lower sum
2is same as Wiener index. In this paper based on the definition of Wiener index we investigate the Wiener index of Fibonacci Labeled Graph P
nʘ F
4.
Keywords: Fibonacci labeling, Wiener Index, Path and Fan Graphs, Carono.
1. INTRODUCTION
A graph labeling is a mapping from graph elements to a nonempty set. We present
those conditions that are necessary and sufficient to whether a graph is Fibonacci labeling of
some graphs in graph theory. The graphs considered here are finite and simple. The vertex set
and edge set of a graph G are denoted by V(G) and E(G) respectively. We take a concept called
Fibonacci labeling some special graphs in graph theory. We defined a new graph called P n ʘ
F 4 in which the edges can be labeled with the first few Fibonacci numbers.
Distance properties of molecular graphs form an important topic in chemical graph theory. To justify this statement just recall the famous Wiener index is also known as the Wiener number. A new approach to the study of distance properties of molecular graphs was proposed by Klavzjar, Gutman, and Mohar. The Wiener index named after the chemist Harold Wiener is one of the most widely known topological descriptor. The Wiener Index has been well studied over the last quarter of a century and it correlates well with many physico chemical properties of organic compounds. The Wiener Index of a graph is defined as the sum of the distances between all vertices pairs in a connected graph.
In mathematical terms a graph is represented as G = (V, E) where V is the set of vertices and E is the set of edges. Let G be an undirected connected graph without loops or multiple edges with n vertices, denoted by 1, 2, . . . , n. The topological distance between a pair of vertices i and j, which is denoted by d(vi, vj ), is the number of edges of the shortest path joining i and j. In 1947 Harold Wiener defined the Wiener index W(G) as the sum of distances between all vertices of the graph G
W(G) =
j i
j i v v d ( , )
The Wiener index W(G) of a graph G is defined as the sum of the half of the distances between every pair of vertices of G.
W(G) =
n
i n
j
j i v v d
1 1
, )
2 ( 1
2. PRELIMINARIES
Definition: 2.1
A graph labeling is the assignment of labels, represented by integers, to the edges or vertices, or both, of a graph. Formally, given a graph G = (V, E), a vertex labeling is a function of V to a set of labels.
Definition: 2.2
Fibonacci numbers can be defined by the linear recurrence relation F n = F n–1 + F n–2 , n≥2, where F 0 = 0, F 1 = 1. This generates the infinite sequence of integers beginning with 0, 1, 1, 2, 3, 5, 8, …
Definition: 2.3
A Graph G will be called Fibonacci labeled if there is a labeling f of its vertices with integers from the set {0, 1, 2, …, F n }, so that the induced edge labeling f* is defined by f*(uv)
= |f(u) – f(v)| is a bijection on to the set {F 0 , F 1 , F 2 , …, F n } Definition: 2.4
Let G = (V(G), E(G)) be a connected undirected graph, any two vertices u, v of V(G),
(u, v) denotes the minimum distance between u and v. Then the Wiener Lower Sum W L (G)
of the graph is defined by
W L (G) = 1 2 ∑ 𝑢,𝑣 ∈ 𝑉 (𝐺) 𝛿(𝑢, 𝑣) where (u, v) = min d(u, v) Definition: 2.5
A path in a graph is a finite or infinite sequence of edges which connect a sequence of vertices which, by most definitions, are all distinct from one another.
Definition: 2.6
A fan graph is defined as the graph join, where is the empty graph on nodes and is the path graph on nodes. The case corresponds to the usual fan graphs, while corresponds to the double fan, etc.
Definition: 2.7
If corona of path and fans are arranged linearly then the P n ʘ F 4 graph is obtained.
Definition: 2.8
The Distance Matrix 4,5 (DM) is a lower or upper triangular matrix whose elements are d(u,v).
Definition: 2.9
Let D(u,v) denote the shortest distance between two vertices u, v V(G). The Wiener index of a graph G with q edges is denoted by W(G:q) = q D(u,v) . The sum is given by D(u,v) = W (G:1) where W denotes the derivative W[G:q] with respect to q.
3. CONSTRUCTION OF P n ʘ F 4 GRAPH
If Fans F 1,4 are arranged linearly with a path then the P n ʘ F 4 graph is obtained. Let The corona G 1 ʘ G 2 of two graphs G 1 (with n 1 vertices and m 1 edges) and G 2 (with n 2 vertices and m 2 edges) is defined as the graph obtained by taking one copy G 1 and n 1 copies of G 2 and then joining the i th vertex of G 1 with an edge to every vertex in the i th copy of G 2 . It follows from the definition of the corona that G 1 ʘG 2 has n 1 + n 1 n 2 vertices and m1+n1m2+n1n2 edges.
It is easy to see that G 1 ʘ G 2 is not in general isomorphic to G 2 ʘ G 1 . The corona is P n ʘ F 4 is Fibonacci labeled and solved wiener index of it.
Fig(i): P
1ʘF
4Fig(ii): P
2ʘF
4Fig(iii): P
nʘF
44. WIENER INDEX OF P n ʘ F 4
Theorem: 3.1
The Wiener Index of P n ʘ F 4 of dimension n 2 is a positive integer such that
6
) 67 120 25
) ( (
2
n
n n n
W .
Proof:
Let G = P n ʘ F 4 , n 2. Let V(G) = {0, 1, 2, …, n} and E(G) = {F 0 , F 1 , F 2 , …, F i , 0 ≤ i ≤ 5} be respectively the vertex set and edge set of G.
The Graph of P 2 ʘF 4 is given in Fig (ii) Consider the distance matrix of P 2 ʘF 4
0 1 2 3 4 5 6 7 8 9
DM =
0 0 1 1 2 2 3 3 2 3 3 1 - 0 1 1 1 2 1 2 2 2 2 - - 0 1 2 3 2 3 3 3 3 - - - 0 1 3 2 3 3 3 4 - - - - 0 3 2 3 3 3 5 - - - - - 0 1 1 2 2 6 - - - - - - 0 1 1 1 7 - - - - - - - 0 1 2 8 - - - - - - - - 0 1 9 - - - - - - - - - 0
W(G:q) = 7q 1 + 3q 2 , where G = P 2 ʘF 4
Similarly,
W(G:q) = 15q 1 + 14q 2 + 16q 3 , where G = P 3 ʘF 4
W(G:q) = q 1 + 31q 2 + 18q 3 + 9q 4 , where G = P 4 ʘF 4
W(G:q) = 25q 1 + 41q 2 + 27q 3 + 18q 4 + 9q 5 , where G = P 5 ʘF 4
W(G:q) = 30q 1 + 51q 2 + 36q 3 + 27q 4 + 18q 5 + 9q 6 , where G = P 6 ʘF 4
…
…
and so on
Wiener Sum = W(1) = 7, W(2) = 32, ….
6
) 67 120 25 ) ( (
2
k
k k k
W
6
] 67 ) 1 ( 120 ) 1 ( 25 )[
1 ) ( 1 (
2
k
k k k
W
6 78 17 ) 75
1 (
2
K
W k
6
78 50 195
25 k
3k
2k
6
] 67 ) 1 ( 120 ) 1 ( 25 )[
1 ) ( 1 (
2
k
k k k
W
By Principle of mathematical induction we can conclude that 6
) 67 120 25
) ( (
2
