On Study of Vertex Labeling of Graph Operations

Vertices of the graphs are labeled from the set of natural numbers from 1 to the order of the given graph. Vertex adjacency label set (AVLS) is the set of ordered pair of vertices and its corresponding label of the graph. A notion of vertex adjacency label number (VALN) is introduced in this paper. For each VLS, VLN of graph is the sum of labels of all the adjacent pairs of the vertices of the graph. , V. Lokesha, and K. M. Niranjan3


Introduction
The vertex natural labeling of graphs is introduced in ref. [1][2][3].Research in the graph enumeration and graph labeling started way back in 1857 by Arthur Cayley.Graph enumeration is defined as counting number of different graphs of particular type, subgraphs, etc. with graph variants (the number of vertices and edges of the graph).Labeling of graph is assigning labels to the vertices or edges of a graph.Most graph labeling concepts trace their origins to labeling presented by Alex Rosa [1].Some of graph labeling methods are graceful labeling, harmonious labeling, and coloring of graphs.For the detailed survey on graph labeling see ref. [4].Graph labeling and enumeration finds the application in chemical graph theory, social networking and computer networking.For example, Cayley demonstrated that the number of different trees of n vertices is analogues to number of isomers of the saturated hydrocarbon with n carbon items C n H 2n+2 extension of our previous work [5], in which vertex labeling for graph operations are studied.The following definitions are used abundantly in this paper.

Vertex natural labeling of a graph:
A vertex natural labeling of G is a mapping function l which assigns each vertex v of G, an unique number ( ) u l from the set of natural numbers { } p N ,......, . That is all the vertices are having distinct labels from 1 to p. Thus l is bijective.So there are !p sets of different labeling of a given graph.Each such label set is called vertex adjacency label set (VALS).Vertex adjacency labeling number (VALN) for each VALS is defined as the sum of labels of all the adjacent pairs of the vertices of the graph, which is given by .

Subdivision of a graph:
The subdivision of graph of a graph G denoted by S(G) is a graph obtained from G by replacing each of its edge by two series edges by introducing a new vertex into each edge of the G. See the ref.[7] for more details.
Direct Sum of two graphs: Direct sum of two graphs ( )

Direct product of two graphs:
The direct product of two graphs ( ) by joining every vertex of ( ) with every vertex of ( ) . Total number of links in complete product of two graphs is This work is an extension of our paper [5] in which ℵ is computed for various graph operation which includes subdivision, direct sum and product operations.The paper is organized into several sections.In section 2, important results and observations which are useful for deriving the new results are described.In section 3, new problems are stated and results are provided for these problems.Conclusions are given in section 4.

Known Results
Most of the graph theory definitions are found in literature [6][7][8].We recall some essential results which are required in a sequel from ref. [5].

i.
For complete graph, p K , any order vertex natural label of graphs gives ℵ .That is all the p! Vertex labeling of graphs yields same ℵ value.ii.
For a ( ) r p G , is r-regular graph, any order vertex natural label of graphs gives ℵ .

iii.
For any graph G, by assigning the labels from the label set { }

Theorem 1:
and the new . Proof: (a).From the data, the subgraph H is obtained from G by removing the vertex on removing the vertex j v from G, results in removal of edges associated from it.This implies the contribution of labels to the summation from the adjacent vertices of j v needs to be removed.Thus Proof: (b).From the above proof, ( ) ( ) . To reduce it to equality, the RHS of the inequality further needs to be subtracted.The vertices which are adjacent to j v can be of two types: vertices with their labels higher or lower than that of j v .In the above inequality, for the adjacent vertices with their labels lower than that of j v only their degree are reduced by 1 and where as for the adjacent vertices with their labels.( .

Proof:
The vertices of ( ) , is a graph obtained by subdivision operation on r-regular graph G, then (a).
Proof: (a).The vertices of ( ) ( ) with vertices of sets V 1 and V 2 r have degree and 2 respectively.
. By substituting these values in result Proof: (b).The vertices of ( ) ( ) , is a graph obtained by subdivision operation on graph G, then )  .Vertices of ( ) ( ) By adding these two, is a graph with p points and q edges.(a).Let ( ) ( ) ) a graph, whose vertices have degree either m or n.Let m p vertices have degree m and n p vertices have degree n.Then

.
Proof: (b).Using the (a), and the degree of the vertex v i Let the set { } , and the corresponding VALS is called maximal vertex adjacency label set(MAVLS).Similarly vertex non adjacency labeling number (VNALN) for each VALS is defined as sum of labels of all non adjacent pairs of the vertices of the graph, which is given by