Schultz and Modified Schultz Polynomials of Vertex Identification Chain for Square and Complete Square Graphs ()
1. Introduction
Given the importance of topological evidence, which can be deduced from the polynomial by finding the derivative with respect to a specific variable and then compensating for this variable by one value, therefore we have in this paper found the polynomial of Schultz and modified Schultz for a chain of special graphs which is the 4-cycle and 4-cycle complete by identification symmetrical vertices.
In this paper, we can refer to the basic concepts in graph theory and topological indices to the references [1] [2] [3]. Let
be a simple connected graph without loop and multi-edges, where
and
be a set of vertices and edges of respect to a graph G. The distance between any two vertices of
is the shortest path between them which denoted its by
,
and the maximum distance between any two vertices in G is called the diameter graph G, that is:
. The degree of vertex u in a graph G is the number of the edges which incident on u and denoted by
.
Introduced Schultz index by Schultz in 1989 [4] and in 1997 Klavžar and Gutman were defined the modified Schultz index [5]. The Schultz index
and the modified Schultz index
are have defined as:
. (1.1)
. (1.2)
The Schultz and modified Schultz polynomials are have defined respectively as:
. (1.3)
. (1.4)
From these polynomials can obtain:
1) The Schultz index:
. (1.5)
2) The modified Schultz index:
. (1.6)
3) The average distance of Schultz:
(1.7)
4) The average distance of modified Schultz:
. (1.8)
where
is represent the derivative w.r.t. x and
is the order of G.
There are many recent papers on polynomials and indices for Schultz and modified Schultz, see to references [6] [7] [8] and there are applications on Schultz and modified Schultz in chemistry, see to references [9] [10] [11].
Let
be the set of all
of G which distance between u and v is k such that
and
and
is the number of pairs
of G that are distance k “
”.
From clearly that
.
2. The Vertex―Identification Chain (VIC)―Graphs
Let
be a set of pairwise disjoint graphs with vertices
,
,
, then the vertex-identification chain graph
of
with respect to the vertices
is the graph obtained from the graphs
by identifying the vertex
with the vertex
for all
. (See Figure 1):
Figure 1. The graph
.
Some Properties of Graph
:
1)
.
2)
.
3)
.
The equality of lower bound is satisfied at
but the upper bound is satisfied at path graph.
If
, for all
, where
is a connected graph of order p, we denoted
by
.
2.1. Schultz and Modified Schultz of
From Figure 2, we note that
,
and
, for all
,
,
,
, then we have:
Table 1. The vertices degrees of the graph
.
Theorem 2.1.1: For
, then we have:
1)
.
2)
.
Proof: For all
and every two vertices
, there is
,
, then obviously,
. We will have
six partitions for proof:
P1. If
, then
and we have two subsets of it:
P1.1.
.
P1.2.
.
P2. If
, then
and we have three subsets of it:
P2.1.
P2.2.
.
P2.3.
.
P3. If
, then
, and we have:
1) If k is an odd,
, then
, and we have two subsets of it:
P3.1.
.
P3.2.
2) If k is an even,
, then
, and we have three
subsets of it:
P3.3.
P3.4.
.
P3.5.
.
P4. If
, then
, and we have two subsets of it:
P4.1.
.
P4.2.
.
P5. If
, then
, and we have one subset of it:
.
P6. If
, then
, and we have one subset of it:
.
Hence, from P1 to P6 and Table 1, we get:
And,
By simple, we can calculate:
1)
.
.
2)
.
. ∎
Corollary 2.1.2: For
, then we have:
1)
.
2)
. ∎
Corollary 2.1.3: For
, then we have:
1)
.
2)
. ∎
2.2. Schultz and Modified Schultz of
From Figure 3, we note that
,
and
, for all
,
,
,
, then we have:
Table 2. The vertices degrees of the graph
.
Theorem 2.2.1: for
, then we have:
1)
.
2)
.
Proof: For all
, and every vertex
there is
,
, then obviously,
. We will have
four partitions for proof:
P1. If
, then
and we have three subsets of it:
P1.1.
P1.2.
P1.3.
.
P2. If
,
, then
and we have three subsets of it:
P2.1
P2.2.
P2.3.
.
P3. If
, then
and we have two subsets of it:
P3.1.
P3.2.
.
P4. If
, then
, and we have one subset of it:
Hence, from P1 to P4 and Table 2, we get:
And,
By simply we can calculate:
.
.
This completes the proof. ∎
Remark:
1)
.
2)
.
Corollary 2.2.2: For
, then we have:
1)
.
2)
. ∎
Corollary 2.2.3: For
, then we have:
1)
.
2)
. ∎
3. Some Properties of the Coefficients of Schultz and Modified Schultz Polynomials
A finite sequence
of h positive integers is coefficients of polynomial
. Then:
1) The polynomial
is called j-unimodal if, for some index j,
and it is strictly j-unimodal if the inequality holds without equalities.
2) The polynomial
is called monotonically increasing (or monotonically decreasing) if,
or
, respectively, for all
and it is strictly-increasing or strictly-decreasing respectively if the inequalities hold without equalities.
3) The polynomial
is called palindromic if
, for all
and is called semi-palindromic if
,
and for all
.
4) The polynomial
is called troubled if
, for all
.
5) The polynomial
is called equality if
, for all
and is called semi-equality if
for some values i.
The following Table 3 shows the properties of polynomials coefficients of Schultz and modified Schultz of
and
.
4. Conclusion
In this paper, we obtained the general formulas of polynomials for Schultz and modified Schultz polynomials of operation vertex identification chain for square and complete square graphs and indices of its. Also, we discuss some properties of the coefficients of these polynomials.
Table 3. The some properties of the coefficients of
and
,
.
Acknowledgements
This paper is has supported from Master’s thesis for the student Mahmood M. A. from college computer sciences and mathematics, University of Mosul, Republic of Iraq.