Schultz and Modified Schultz Polynomials of Some Cog-Special Graphs ()
1. Introduction
We follow the terminology of [1] [2] [3] [4] . All the graphs considered in this paper are simple and connected finite undirected without loops or multiple edges. Distance is an important concept in graph theory and it has applications to computer science, chemistry, and a variety of other fields [5] [6] .
Suppose that
is a simple undirected connected graph of order
and size
, the distance between two vertices u and v of G is denoted by
and it is defined as the length of a shortest
-path in connected graph G. In particular, if
, then
. The greatest distance in G is the diameter and will be denoted by D. The number of pairs of vertices of G that are distance k is denoted by
. Let
be the set of all unordered pairs of vertices with distance k such that
and
, where
is representation of the number of unordered pairs distinct vertices in G.
The Schultz polynomial of a graph G is defined as:
,
and modified Schultz polynomial of a graph G is defined as:
.
The molecular topological index (Schultz index) was introduced by Harry P. Schultz in 1993 [7] and the modified Schultz index was defined by S. Klavžar and I. Gutman in 1997 [8] .
The Schultz index is defined as:
,
and modified Schultz index is defined as:
.
where the summation for all above is taken over all unordered pairs of distinct vertices in V(G).
The indices of Schultz and modified Schultz can be obtained by the derivative of Schultz and modified Schultz polynomials with respect to x at
, i.e.:
, and
respectively.
The average distance of a connected graph G with respect Schultz and modified Schultz is defined as:
and
.
Schultz and modified Schultz polynomial of two operations Gutman’s and the Cog-complete bipartite Graphs founded by Ahmed and Haitham [9] [10] , the Schultz and modified Schultz polynomial of some special graphs are summarized in the following theorem (See [11] ).
Theorem 1.1:
1)
,
.
2)
,
.
3)
,
.
4)
,
.
5)
2. Main Results
2.1. Definition
A cog-complete graph
is the graph constructed from a complete graph
,
, of vertex set
with p additional vertices
, and 2p edges
,
, as shown in Figure 1.
It is clear that
,
, and
, for
.
Theorem 2.1.1: For
, we have:
1)
2)
Proof: For every vertice
, there is
,
, and obviously
.
We will have three partitions for proof:
P1. If
, then
and is equal to
, we have two subsets of it:
P1.1.
P1.2.
P2. If
, then
, we have two subsets of it:
P2.1.
.
P2.2.
P3. If
, then
, we have:
From P1 - P3, we have:
Corollary 2.1.2: For
, we have:
1)
.
2)
.
Corollary 2.1.3: For
, we have:
1)
.
2)
.
Remark 2.1.4:
1)
.
2)
.
2.2. Definition
A cog-star graph
is the graph constructed from a star graph,
,
, of vertex set
with p additional vertices
, and edges
,
, as shown in Figure 2.
It is clear that
,
,
for
.
Theorem 2.2.1: For
, we have:
1)
2)
Proof: For every vertice
, there is
,
, and obviously
.
We will have four partitions for proof:
P1. If
, then
and is equal to
, we have two subsets of it:
P1.1.
P1.2.
P2. If
, then
, we have three subsets:
P2.1.
P2.2.
P2.3.
P3. If
, then
, we have:
P4. If
, then
, we have:
From P1 - P4, we have:
Corollary 2.2.2: For
, we have:
1)
.
2)
.
Corollary 2.2.3: For
, we have:
1)
.
2)
.
Remark 2.2.3:
1)
.
2)
.
2.3. Definition
A cog-wheel graph
is the graph constructed from a wheel
,
, of order
, with vertex set
and with p additional vertices
, and edges
,
, as shown in Figure 3.
It is clear that
,
,
for
.
Theorem 2.3.1: For
, we have:
1)
2)
Proof: For every vertice
, there is
,
, and obviously
.
We will have four partitions for proof:
P1. If
, then
and is equal to
, we have three subsets of it:
P1.1.
P1.2.
P1.3.
P2. If
, then
, we have five subsets
P2.1.
P2.2.
.
P2.3.
P2.4.
P2.5.
P3. If
, then
, we have two subsets:
P3.1.
P3.2.
P4. If
, then
, we have:
From P1 - P4, we have:
Corollary 2.3.2: For
, we have:
1)
.
2)
.
Corollary 2.3.3: For
, we have:
1)
.
2)
.
Remark 2.3.4:
1)
,
,
.
2)
,
,
.
2.4. Definition
A saw graph
is a path of order p, say
, with
additional vertices
and edges
as depicted in Figure 4.
It is clear
,
and
, for
.
Theorem 2.4.1: For
, we have:
1)
2)
Proof:
For every vertice
, there is
,
, and obviously
.
We will have four partitions for proof:
P1. if
, then
and is equal to
, we have five subsets of it:
P1.1.
P1.2.
P1.3.
P1.4.
P1.5.
P2. if
,
, then
, we have six subsets of it:
P2.1.
P2.2.
P2.3.
P2.4.
P2.5.
P2.6.
P3. if
then
, we have six subsets of it:
P3.1.
P3.2.
P3.3.
P3.4.
P3.5.
P3.6.
P4. if
then
, we have four subsets of it:
P4.1.
P4.2.
P4.3.
P4.4.
From P1 - P4, we have:
Corollary2.4.2: For
, then:
1)
.
2)
.
Corollary2.4.3: For
, then:
1)
.
2)
.
Remark 2.4.4:
1)
,
.
2)
,
.
2.5. Definition
A Cog-Cycle is a graph
obtained from a cycle graph
with p additional vertices
, and edges
as shown in Figure 5.
It’s clear that
, and
Theorem 2.5.1: For
, then:
1)
2)
Proof: For every vertice
, there is
,
, and obviously
. We will four partitions for proof:
P1. if
, then
and is equal to
. We have three subsets of it:
P1.1.
P1.2.
P1.3.
P2. If
,
, then
. We have four subsets of it:
when
moving to
clockwise.
P2.1.
P2.2.
when
moving to
reversed clockwise.
P2.3.
P2.4.
P3. If
, when p is even, then
, we have four subsets of it:
P3.1.
when
moving to
clockwise.
P3.2.
when
moving to
reversed clockwise.
P3.3.
P3.4.
when p is odd, then
, we have four subsets of it:
P’3.1.
when
moving to
clockwise.
P’3.2.
when
moving to
reversed clockwise.
P’3.3.
P’3.4.
P4. If
, when p is even then
, we have:
when p is odd then
, we have two subsets of it:
P4.1.
P4.2.
From P1 - P4, we have:
Corollary2.5.2: For
, then:
1)
.
2)
Corollary2.5.3: For
, then:
1)
.
2)
.
Remark 2.5.4:
1)
,
.
2)
,
.