A Note on SK, SK 1 , SK 2 Indices of Interval Weighted Graphs

In this study, the SK, SK 1 and SK 2 indices are defined on weighted graphs. Then, the SK, SK 1 and SK 2 indices are defined on interval weighted graphs. Their behaviors are investigated under some graph operations by using these definitions.


Introduction
A topological index of a chemical compound is an integer, derived following a certain rule, which can be used to characterize the chemical compound and predict certain physiochemical properties like boiling point, molecular weight, density, refractive index, and so forth [1].
Molecules and molecular compounds are often modeled by molecular graph.
A molecular graph is a representation of the structural formula of a chemical compound in terms of graph theory, whose vertices correspond to the atoms of the compound and edges correspond to chemical bonds [2]. is the number of first neighbors of v. The edge of the graph G, connecting the vertices u and v, will be denoted by e uv = . Throughout this paper, the graphs considered are assumed to be connected. A connected graph is a graph such that there is a path between all pairs of vertices, see books [3] [4].
We now recall some graph operations we shall need in this paper. , be two simple graphs. The sum 1 2 G G + of these two graphs is defined as the graph having the vertex set ( ) The cartesian product 1 2 G G × is the graph with vertex set ( )

Graph Operations on the SK, SK1, SK2 Indices of Weighted Graphs
In this section, we define the SK, SK 1 and SK 2 indices on weighted graphs. A weighted graph is a graph each edge of which has been assigned to a number called the weight of the edge. All the weight of the edges are assumed to be posi- w u is the sum of the weights on u.
be a connected weighted graph having n vertices. Let each edge of G be weighted with positive real numbers. The weighted G w of G is defined as follows: w u is the sum of the weights on u.
be two simple, connected graphs. Then the SK, SK 1 and SK 2 indices of the sum of graphs 1 G and 2 G are respectively given by be two simple, connected weighted graphs. Then the weighted SK, SK 1 and SK 2 indices of the sum of graphs 1 G and 2 G are respectively given by be two simple, connected graphs. Then the SK, SK 1 and SK 2 indices of the cartesian product of graphs 1 G and 2 G are respectively given by

Graph Operations on the SK, SK1, SK2 Indices of Interval Weighted Graphs
In this section, we define the SK, SK 1 and SK 2 indices on interval weighted graphs. An interval weighted graph (interval graph) is a weighted graph in which each edge is assigned an interval or an interval square matrix. All the interval square matrices are assumed to be of the same order and to be positive definite [8].
Let G be an interval graph on n vertices. Denote by ij w  the positive definite is the sum of the interval weights on u. G w  of G is defined as follows: is the sum of the interval weights on u. G w  of G is defined as follows: is the sum of the interval weights on u.
For the second sum 2 K , we take the vertices i and j in ( ) 2 V G so that ij is in In the third sum 3 K , i is taken in ( ) 1 V G and j is taken in ( ) The result now follows by adding the three contributions and simplifying the resulting expression.