Schultz and Modified Schultz Polynomials of Some Cog-Special Graphs

For a connected graph G, the Schultz and modified Schultz polynomials are defined as, respectively, where the summations are taken over all unordered pairs of distinct vertices in V(G), is the degree of vertex u, is the distance between u and v and V(G) is the vertex set of G. In this paper, we find Schultz and modified Schultz polynomials of the Cog-special graphs such as a complete graph, a star graph, a wheel graph, a path graph and a cycle graph. The Schultz index, modified Schultz index and average distance of Schultz and modified Schultz of each such Cog-special graphs are also obtained in this paper.

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Ali, A. and Mohammed, H. (2019) Schultz and Modified Schultz Polynomials of Some Cog-Special Graphs. Open Access Library Journal, 6, 1-13. doi: 10.4236/oalib.1105625. 1. Introduction

We follow the terminology of     . 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   .

Suppose that $G=\left(V\left(G\right),E\left(G\right)\right)$ is a simple undirected connected graph of order $p=p\left(G\right)=|V\left(G\right)|$ and size $q=q\left(G\right)=|E\left(G\right)|$ , the distance between two vertices u and v of G is denoted by $d\left(u,v\right)$ and it is defined as the length of a shortest $\left(u,v\right)$ -path in connected graph G. In particular, if $u=v$ , then $d\left(u,v\right)=0$ . 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 $d\left(G,k\right)$ . Let ${D}_{k}\left(G\right)$ be the set of all unordered pairs of vertices with distance k such that $|{D}_{k}\left(G\right)|=d\left(G,k\right)$ and $\underset{k=1}{\overset{D}{\sum }}d\left(G,k\right)=\left(\begin{array}{l}p\\ 2\end{array}\right)$ , where $\left(\begin{array}{l}p\\ 2\end{array}\right)$ is representation of the number of unordered pairs distinct vertices in G.

The Schultz polynomial of a graph G is defined as:

$Sc\left(G;x\right)=\underset{u,v\in V\left(G\right)}{\sum }\left({\delta }_{u}+{\delta }_{v}\right){x}^{d\left(u,v\right)}$ ,

and modified Schultz polynomial of a graph G is defined as:

$\stackrel{*}{Sc}\left(G;x\right)=\underset{u,v\in V\left(G\right)}{\sum }\left({\delta }_{u}{\delta }_{v}\right){x}^{d\left(u,v\right)}$ .

The molecular topological index (Schultz index) was introduced by Harry P. Schultz in 1993  and the modified Schultz index was defined by S. Klavžar and I. Gutman in 1997  .

The Schultz index is defined as:

$Sc\left(G\right)=\underset{u,v\in V\left(G\right)}{\sum }\left({\delta }_{u}+{\delta }_{v}\right)d\left(u,v\right)$ ,

and modified Schultz index is defined as:

$\stackrel{*}{Sc}\left(G\right)=\underset{u,v\in V\left(G\right)}{\sum }\left({\delta }_{u}{\delta }_{v}\right)d\left(u,v\right)$ .

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 $x=1$ , i.e.:

$Sc\left(G\right)={\frac{\text{d}}{\text{d}x}Sc\left(G;x\right)|}_{x=1}$, and $\stackrel{*}{Sc}\left(G\right)={\frac{\text{d}}{\text{d}x}\stackrel{*}{Sc}\left(G;x\right)|}_{x=1}$ respectively.

The average distance of a connected graph G with respect Schultz and modified Schultz is defined as:

$\stackrel{¯}{Sc\left(G\right)}=\frac{Sc\left(G\right)}{\left(\begin{array}{l}p\\ 2\end{array}\right)}$ and $\stackrel{¯}{\stackrel{*}{Sc}\left(G\right)}=\frac{\stackrel{*}{Sc}\left(G\right)}{\left(\begin{array}{l}p\\ 2\end{array}\right)}$ .

Schultz and modified Schultz polynomial of two operations Gutman’s and the Cog-complete bipartite Graphs founded by Ahmed and Haitham   , the Schultz and modified Schultz polynomial of some special graphs are summarized in the following theorem (See  ).

Theorem 1.1:

1) $Sc\left({K}_{p};x\right)=p{\left(p-1\right)}^{2}{x}^{1}$ , $\stackrel{*}{Sc}\left({K}_{p};x\right)=\left\{p{\left(p-1\right)}^{3}/2\right\}{x}^{1}$ .

2) $Sc\left({S}_{p+1};x\right)=p\left(p+1\right){x}^{1}+p\left(p-1\right){x}^{2}$ , $\stackrel{*}{Sc}\left({S}_{p+1};x\right)={p}^{2}{x}^{1}+\left\{p\left(p-1\right)/2\right\}{x}^{2}$ .

3) $Sc\left({W}_{p+1};x\right)=\left({p}^{2}+9p+6\right){x}^{1}+3p\left(p-3\right){x}^{2}$ , $\stackrel{*}{Sc}\left({W}_{p+1};x\right)=3\left({p}^{2}+3p+3\right){x}^{1}+\left\{9p\left(p-3\right)/2\right\}{x}^{2}$ .

4) $Sc\left({P}_{p};x\right)=\underset{k=1}{\overset{p-1}{\sum }}\left[4\left(p-k\right)-2\right]{x}^{k}$ , $\stackrel{*}{Sc}\left({P}_{p};x\right)=4\underset{k=1}{\overset{p-1}{\sum }}\left(p-k-1\right){x}^{k}+{x}^{p-1}$ .

5) $Sc\left({C}_{p};x\right)=\stackrel{*}{Sc}\left({C}_{p};x\right)=4p\underset{k=1}{\overset{⌈p/2⌉-1}{\sum }}{x}^{k}+\left[\begin{array}{l}2p{x}^{p/2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}p\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{even},\\ 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}p\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{odd}.\end{array}$

2. Main Results

2.1. Definition

A cog-complete graph ${K}_{p}^{c}$ is the graph constructed from a complete graph ${K}_{p}$ , $p\ge 3$ , of vertex set $\left\{{u}_{1},{u}_{2},\cdots ,{u}_{p}\right\}$ with p additional vertices $\left\{{v}_{1},{v}_{2},\cdots ,{v}_{p}\right\}$ , and 2p edges $\left\{{v}_{i}{u}_{i},{v}_{i}{u}_{i+1}:i=1,2,\cdots ,p\right\}$ , $\left({u}_{p+1}\equiv {u}_{1}\right)$ , as shown in Figure 1.

It is clear that $p\left({K}_{p}^{c}\right)=2p$ , $q\left({K}_{p}^{c}\right)=p\left(p+3\right)/2$ , and $diam\text{\hspace{0.17em}}{K}_{p}^{c}=3$ , for $p\ge 4$ .

Theorem 2.1.1: For $p\ge 4$ , we have:

1) $Sc\left({K}_{p}^{c};x\right)=p\left({p}^{2}+2p+5\right){x}^{1}+p\left(p-1\right)\left(p+2\right){x}^{2}+2p\left(p-3\right){x}^{3}.$

2) $\stackrel{\ast }{Sc}\left({K}_{p}^{c};x\right)=\left\{p\left(p+1\right)\left({p}^{2}+7\right)/2\right\}{x}^{1}+2{p}^{2}\left(p-1\right){x}^{2}+2p\left(p-3\right){x}^{3}.$

Proof: For every vertice $y,z\in V\left({K}_{p}^{c}\right)$ , there is $d\left(y,z\right)=k$ , $k=1,2,3$ , and obviously $\underset{i=1}{\overset{3}{\sum }}|{D}_{i}|=p\left(2p-1\right)$ .

We will have three partitions for proof:

P1. If $d\left(y,z\right)=1$ , then $|{D}_{1}|=p\left(p+3\right)/2$ and is equal to $q\left({K}_{p}^{c}\right)$ , we have two subsets of it:

P1.1. $\begin{array}{l}|\left\{\left({u}_{i},{v}_{j}\right):{u}_{i}{v}_{j}\in E\left({K}_{p}^{c}\right),{\delta }_{{u}_{i}}+{\delta }_{{v}_{j}}=p+3&{\delta }_{{u}_{i}}{\delta }_{{v}_{j}}=2\left(p+1\right),\\ i=j,j+1,1\le j\le p,\left({u}_{p+1}\equiv {u}_{1}\right)\right\}|=2p.\end{array}$

P1.2. $\begin{array}{l}|\left\{\left({u}_{i},{u}_{j}\right):{u}_{i}{u}_{j}\in E\left({K}_{p}^{c}\right),{\delta }_{{u}_{i}}+{\delta }_{{u}_{j}}=2\left(p+1\right)\\ \text{ }&{\delta }_{{u}_{i}}{\delta }_{{u}_{j}}={\left(p+1\right)}^{2},1\le i,j\le p,i\ne j\right\}|=p\left(p-1\right)/2.\end{array}$

Figure 1. A cog-complete graph ${K}_{p}^{c}$ .

P2. If $d\left(y,z\right)=2$ , then $|\text{\hspace{0.17em}}{D}_{2}|=p\left(p-1\right)$ , we have two subsets of it:

P2.1. $|\left\{\left({v}_{i},{v}_{i+1}\right):{\delta }_{{v}_{i}}+{\delta }_{{v}_{i+1}}=4&{\delta }_{{v}_{i}}{\delta }_{{v}_{i+1}}=4,1\le i\le p,\left({v}_{p+1}\equiv {v}_{1}\right)\right\}|=p$ .

P2.2. $\begin{array}{l}|\left\{\left({v}_{i},{u}_{j}\right):{v}_{i}{u}_{j}\notin E\left({K}_{p}^{c}\right),{\delta }_{{v}_{i}}+{\delta }_{{u}_{j}}=p+3&{\delta }_{{v}_{i}}{\delta }_{{u}_{j}}=2\left(p+1\right),\\ 1\le i,j\le p,i\ne j,j+1\left({u}_{p+1}\equiv {u}_{1}\right)\right\}|=p\left(p-2\right).\end{array}$

P3. If $d\left(y,z\right)=3$ , then $|{D}_{3}|=p\left(p-3\right)/2$ , we have:

$\begin{array}{l}|\left\{\left({v}_{i},{v}_{j}\right):{\delta }_{{v}_{i}}+{\delta }_{{v}_{j}}=4&{\delta }_{{v}_{i}}{\delta }_{{v}_{j}}=4,1\le i,j\le p,|i-j|\ne 0,1\right\}|-|\left\{\left({v}_{1},{v}_{p}\right)\right\}|\\ =p\left(p-3\right)/2.\end{array}$

From P1 - P3, we have:

$Sc\left({K}_{p}^{c};x\right)=p\left({p}^{2}+2p+5\right){x}^{1}+p\left(p-1\right)\left(p+2\right){x}^{2}+2p\left(p-3\right){x}^{3}.$

$\stackrel{*}{Sc}\left({K}_{p}^{c};x\right)=\left\{p\left(p+1\right)\left({p}^{2}+7\right)/2\right\}{x}^{1}+2{p}^{2}\left(p-1\right){x}^{2}+2p\left(p-3\right){x}^{3}.$

Corollary 2.1.2: For $p\ge 4$ , we have:

1) $Sc\left({K}_{p}^{c}\right)=p\left(3{p}^{2}+10p-17\right)$ .

2) $\stackrel{\ast }{Sc}\left({K}_{p}^{c}\right)=p\left({p}^{3}+9{p}^{2}+11p-29\right)/2$ .

Corollary 2.1.3: For $p\ge 4$ , we have:

1) $10\frac{1}{7}\le \stackrel{¯}{Sc}\left({K}_{p}^{c}\right)<\left(6p+23\right)/4$ .

2) $15\frac{13}{14}\le \stackrel{¯}{\stackrel{\ast }{Sc}}\left({K}_{p}^{c}\right)<\left(4{p}^{2}+38p+63\right)/16$ .

Remark 2.1.4:

1) $Sc\left({K}_{3}^{c};x\right)=60{x}^{1}+30{x}^{2}$ .

2) $\stackrel{\ast }{Sc}\left({K}_{3}^{c};x\right)=96{x}^{1}+36{x}^{2}$ .

2.2. Definition

A cog-star graph ${S}_{p+1}^{c}$ is the graph constructed from a star graph, ${S}_{p+1}$ , $p\ge 3$ , of vertex set $\left\{{u}_{0},{u}_{1},\cdots ,{u}_{p-1},{u}_{p}\right\}$ with p additional vertices $\left\{{v}_{1},{v}_{2},\cdots ,{v}_{p-1},{v}_{p}\right\}$ , and edges $\left\{{v}_{i}{u}_{i},{v}_{i}{u}_{i+1}:i=1,2,\cdots ,p\right\}$ , $\left({u}_{p+1}\equiv {u}_{1}\right)$ , as shown in Figure 2.

Figure 2. A cog-star graph ${S}_{p+1}^{c}$ .

It is clear that $p\left({S}_{p+1}^{c}\right)=2p+1$ , $q\left({S}_{p+1}^{c}\right)=3p$ , $diam\text{\hspace{0.17em}}{S}_{p+1}^{c}=4$ for $p\ge 4$ .

Theorem 2.2.1: For $p\ge 4$ , we have:

1) $Sc\left({S}_{p+1}^{c};x\right)=p\left(p+13\right){x}^{1}+p\left(4p+3\right){x}^{2}+5p\left(p-2\right){x}^{3}+2p\left(p-3\right){x}^{4}.$

2) $\stackrel{\ast }{Sc}\left({S}_{p+1}^{c};x\right)=3p\left(p+4\right){x}^{1}+\left\{p\left(13p-1\right)/2\right\}{x}^{2}+6p\left(p-2\right){x}^{3}+2p\left(p-3\right){x}^{4}$

Proof: For every vertice $y,z\in V\left({S}_{p+1}^{c}\right)$ , there is $d\left(y,z\right)=k$ , $k=1,2,3,4$ , and obviously $\underset{i=1}{\overset{4}{\sum }}|{D}_{i}|=p\left(2p+1\right)$ .

We will have four partitions for proof:

P1. If $d\left(y,z\right)=1$ , then $|{D}_{1}|=3p$ and is equal to $q\left({S}_{p+1}^{c}\right)$ , we have two subsets of it:

P1.1. $|\left\{\left({u}_{0},{u}_{i}\right):{u}_{0}{u}_{i}\in E\left({S}_{p+1}^{c}\right),{\delta }_{{u}_{0}}+{\delta }_{{u}_{i}}=p+3&{\delta }_{{u}_{0}}{\delta }_{{u}_{i}}=3p,1\le i\le p\right\}|=p.$

P1.2. $\begin{array}{l}|\left\{\left({v}_{i},{u}_{j}\right):{v}_{i}{u}_{j}\in E\left({S}_{p+1}^{c}\right),{\delta }_{{v}_{i}}+{\delta }_{{u}_{j}}=5&{\delta }_{{v}_{i}}{\delta }_{{u}_{j}}=6,1\le i\le p,\\ j=i,i+1,\left({u}_{p+1}\equiv {u}_{1}\right)\right\}|=2p.\end{array}$

P2. If $d\left(y,z\right)=2$ , then $|{D}_{2}|=p\left(p+3\right)/2$ , we have three subsets:

P2.1. $|\left\{\left({u}_{i},{u}_{j}\right):{\delta }_{{u}_{i}}+{\delta }_{{u}_{j}}=6&{\delta }_{{u}_{i}}{\delta }_{{u}_{j}}=9,1\le i,j\le p,i\ne j\right\}|=p\left(p-1\right)/2.$

P2.2. $|\left\{\left({u}_{0},{v}_{i}\right):{\delta }_{{u}_{0}}+{\delta }_{{v}_{i}}=p+2&{\delta }_{{u}_{0}}{\delta }_{{v}_{i}}=2p,1\le i\le p\right\}|=p.$

P2.3. $|\left\{\left({v}_{i},{v}_{i+1}\right):{\delta }_{{v}_{i}}+{\delta }_{{v}_{i+1}}=4&{\delta }_{{v}_{i}}{\delta }_{{v}_{i+1}}=4,1\le i\le p,\left({v}_{p+1}\equiv {v}_{1}\right)\right\}|=p.$

P3. If $d\left(y,z\right)=3$ , then $|{D}_{3}|=p\left(p-2\right)$ , we have:

$\begin{array}{l}|\left\{\left({u}_{i},{v}_{j}\right):{\delta }_{{u}_{i}}+{\delta }_{{v}_{j}}=5&{\delta }_{{u}_{i}}{\delta }_{{v}_{j}}=6,1\le i,j\le p,i-j\ne 0,1\right\}-\left\{\left({u}_{1},{v}_{p}\right)\right\}|\\ =p\left(p-2\right).\end{array}$

P4. If $d\left(y,z\right)=4$ , then $|{D}_{4}|=p\left(p-3\right)/2$ , we have:

$\begin{array}{l}|\left\{\left({v}_{i},{v}_{j}\right):{\delta }_{{v}_{i}}+{\delta }_{{v}_{j}}=4&{\delta }_{{v}_{i}}{\delta }_{{v}_{j}}=4,1\le i\le p-2,i+2\le j\le p\right\}-\left\{\left({v}_{1},{v}_{p}\right)\right\}|\\ =p\left(p-3\right)/2.\end{array}$

From P1 - P4, we have:

$Sc\left({S}_{p+1}^{c};x\right)=p\left(p+13\right){x}^{1}+p\left(4p+3\right){x}^{2}+5p\left(p-2\right){x}^{3}+2p\left(p-3\right){x}^{4}.$

$\stackrel{*}{Sc}\left({S}_{p+1}^{c};x\right)=3p\left(p+4\right){x}^{1}+\left\{p\left(13p-1\right)/2\right\}{x}^{2}+6p\left(p-2\right){x}^{3}+2p\left(p-3\right){x}^{4}.$

Corollary 2.2.2: For $p\ge 4$ , we have:

1) $Sc\left({S}_{p+1}^{c}\right)=p\left(32p-35\right)$ .

2) $\stackrel{\ast }{Sc}\left({S}_{p+1}^{c}\right)=7p\left(6p-7\right)$ .

Corollary 2.2.3: For $p\ge 4$ , we have:

1) $10\frac{1}{3}\le \stackrel{¯}{Sc}\left({S}_{p+1}^{c}\right)<16$ .

2) $13\frac{2}{9}\le \stackrel{¯}{\stackrel{\ast }{Sc}}\left({S}_{p+1}^{c}\right)<21$ .

Remark 2.2.3:

1) $Sc\left({S}_{4}^{c};x\right)=48{x}^{1}+45{x}^{2}+15{x}^{3}$ .

2) $\stackrel{\ast }{Sc}\left({S}_{4}^{c};x\right)=63{x}^{1}+57{x}^{2}+18{x}^{3}$ .

2.3. Definition

A cog-wheel graph ${W}_{p+1}^{c}$ is the graph constructed from a wheel ${W}_{p+1}$ , $p\ge 3$ , of order $p+1$ , with vertex set $\left\{{u}_{0},{u}_{1},{u}_{2},\cdots ,{u}_{p}\right\}$ and with p additional vertices ${v}_{1},{v}_{2},\cdots ,{v}_{p}$ , and edges $\left\{{v}_{i}{u}_{i},{v}_{i}{u}_{i+1}:i=1,2,\cdots ,p\right\}$ , $\left({u}_{p+1}\equiv {u}_{1}\right)$ , as shown in Figure 3.

It is clear that $p\left({W}_{p+1}^{c}\right)=2p+1$ , $q\left({W}_{p+1}^{c}\right)=4p$ , $diam\text{\hspace{0.17em}}{W}_{p+1}^{c}=4$ for $p\ge 6$ .

Theorem 2.3.1: For $p\ge 6$ , we have:

1) $Sc\left({W}_{p+1}^{c};x\right)=p\left(p+29\right){x}^{1}+p\left(6p+5\right){x}^{2}+p\left(7p-24\right){x}^{3}+2p\left(p-5\right){x}^{4}.$

2) $\begin{array}{l}\stackrel{\ast }{Sc}\left({W}_{p+1}^{c};x\right)=5p\left(p+9\right){x}^{1}+\left\{p\left(29p-27\right)/2\right\}/{x}^{2}+2p\left(5p-18\right){x}^{3}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+2p\left(p-5\right){x}^{4}.\end{array}$

Proof: For every vertice $y,z\in V\left({W}_{p+1}^{c}\right)$ , there is $d\left(y,z\right)=k$ , $k=1,2,3,4$ , and obviously $\underset{i=1}{\overset{4}{\sum }}|{D}_{i}|=p\left(2p+1\right)$ .

We will have four partitions for proof:

P1. If $d\left(y,z\right)=1$ , then $|{D}_{1}|=4p$ and is equal to $q\left({W}_{p+1}^{c}\right)$ , we have three subsets of it:

P1.1. $|\left\{\left({u}_{0},{u}_{i}\right):{u}_{0}{u}_{i}\in E\left({W}_{p+1}^{c}\right),{\delta }_{{u}_{0}}+{\delta }_{{u}_{i}}=p+5&{\delta }_{{u}_{0}}{\delta }_{{u}_{i}}=5p,1\le i\le p\right\}|=p.$

P1.2. $\begin{array}{l}|\left\{\left({u}_{i},{u}_{i+1}\right):{u}_{i}{u}_{i+1}\in E\left({W}_{p+1}^{c}\right),{\delta }_{{u}_{i}}+{\delta }_{{u}_{i+1}}=10\\ \text{ }&{\delta }_{{u}_{i}}{\delta }_{{u}_{i+1}}=25,1\le i\le p,\left({u}_{p+1}\equiv {u}_{1}\right)\right\}|=p.\end{array}$

P1.3. $\begin{array}{l}|\left\{\left({v}_{i},{u}_{j}\right):{v}_{i}{u}_{j}\in E\left({W}_{p+1}^{c}\right),{\delta }_{{v}_{i}}+{\delta }_{{u}_{j}}=7\\ \text{ }&{\delta }_{{v}_{i}}{\delta }_{{u}_{j}}=10,1\le i\le p,j=i,i+1,\left({u}_{p+1}\equiv {u}_{1}\right)\right\}|=2p.\end{array}$

P2. If $d\left(y,z\right)=2$ , then $|{D}_{2}|=p\left(p+5\right)/2$ , we have five subsets

P2.1. $\begin{array}{l}|\left\{\left({u}_{i},{u}_{j}\right):{\delta }_{{u}_{i}}+{\delta }_{{u}_{j}}=10&{\delta }_{{u}_{i}}{\delta }_{{u}_{j}}=25,1\le i\le p-2,i+2\le j\le p\right\}\\ -\left\{\left({u}_{1},{u}_{p}\right)\right\}|=p\left(p-3\right)/2.\end{array}$

Figure 3. A cog-wheel graph ${W}_{p+1}^{c}$ .

P2.2. $|\left\{\left({u}_{0},{v}_{i}\right):{\delta }_{{u}_{0}}+{\delta }_{{v}_{i}}=p+2&{\delta }_{{u}_{0}}{\delta }_{{v}_{i}}=2p,1\le i\le p\right\}|=p$ .

P2.3. $|\left\{\left({u}_{i},{v}_{i+1}\right):{\delta }_{{u}_{i}}+{\delta }_{{v}_{i+1}}=7&{\delta }_{{u}_{i}}{\delta }_{{v}_{i+1}}=10,1\le i\le p,\left({v}_{p+1}\equiv {v}_{1}\right)\right\}|=p.$

P2.4. $\begin{array}{l}|\left\{\left({u}_{i},{v}_{i-2}\right):{\delta }_{{u}_{i}}+{\delta }_{{v}_{i-2}}=7&{\delta }_{{u}_{i}}{\delta }_{{v}_{i-2}}=10,3\le i\le p\right\}\\ \cup \left\{\left({u}_{1},{v}_{p-1}\right),\left({u}_{2},{v}_{p}\right)\right\}|=p.\end{array}$

P2.5. $|\left\{\left({v}_{i},{v}_{i+1}\right):{\delta }_{{v}_{i}}+{\delta }_{{v}_{i+1}}=4&{\delta }_{{v}_{i}}{\delta }_{{v}_{i+1}}=4,1\le i\le p,\left({v}_{p+1}\equiv {v}_{1}\right)\right\}|=p.$

P3. If $d\left(y,z\right)=3$ , then $|{D}_{3}|=p\left(p-3\right)$ , we have two subsets:

P3.1. $\begin{array}{l}|\left\{\left({u}_{i},{v}_{j}\right):{\delta }_{{u}_{i}}+{\delta }_{{v}_{j}}=7&{\delta }_{{u}_{i}}{\delta }_{{v}_{j}}=10,3\le i\le p,1\le j\le p,j\ne i-2,i-1,\\ i,i+1,\left({v}_{p+1}\equiv {v}_{1}\right)\right\}\cup \left\{\left({u}_{i},{v}_{j}\right):i=1,2,i+2\le j\le p+i-3\right\}|\\ =p\left(p-4\right).\end{array}$

P3.2.

$|\left\{\left({v}_{i},{v}_{i+2}\right):{\delta }_{{v}_{i}}+{\delta }_{{v}_{i+2}}=4&{\delta }_{{v}_{i}}{\delta }_{{v}_{i+2}}=4,1\le i\le p,\left({v}_{p+1}\equiv {v}_{1}\right),\left({v}_{p+2}\equiv {v}_{2}\right)\right\}|=p.$

P4. If $d\left(y,z\right)=4$ , then $|{D}_{4}|=p\left(p-5\right)/2$ , we have:

$\begin{array}{l}|\left\{\left({v}_{i},{v}_{j}\right):{\delta }_{{v}_{i}}+{\delta }_{{v}_{j}}=4&{\delta }_{{v}_{i}}{\delta }_{{v}_{j}}=4,3\le i\le p,1\le j\le p,j\ne i-2,i-1,i,i+1,\\ i+2,\left({v}_{p+1}\equiv {v}_{1}\right),\left({v}_{p+2}\equiv {v}_{2}\right)\right\}\cup \left\{\left({v}_{i},{v}_{j}\right):i=1,2,i+3\le j\le p+i-3\right\}|\\ =p\left(p-5\right)/2.\end{array}$

From P1 - P4, we have:

$Sc\left({W}_{p+1}^{c};x\right)=p\left(p+29\right){x}^{1}+p\left(6p+5\right){x}^{2}+p\left(7p-24\right){x}^{3}+2p\left(p-5\right){x}^{4}.$

$\begin{array}{l}\stackrel{*}{Sc}\left({W}_{p+1}^{c};x\right)=5p\left(p+9\right){x}^{1}+\left\{p\left(29p-27\right)/2\right\}{x}^{2}+2p\left(5p-18\right){x}^{3}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+2p\left(p-5\right){x}^{4}.\end{array}$

Corollary 2.3.2: For $p\ge 6$ , we have:

1) $Sc\left({W}_{p+1}^{c}\right)=p\left(42p-73\right)$ .

2) $\stackrel{\ast }{Sc}\left({W}_{p+1}^{c}\right)=2p\left(36p-65\right)$ .

Corollary 2.3.3: For $p\ge 6$ , we have:

1) $13\frac{10}{13}\le \stackrel{¯}{Sc}\left({W}_{p+1}^{c}\right)<21$ .

2) $23\frac{3}{13}\le \stackrel{¯}{\stackrel{\ast }{Sc}}\left({W}_{p+1}^{c}\right)<36$ .

Remark 2.3.4:

1) $Sc\left({W}_{6}^{c};x\right)=170{x}^{1}+175{x}^{2}+55{x}^{3}$ , $Sc\left({W}_{5}^{c};x\right)=132{x}^{1}+116{x}^{2}+8{x}^{3}$ , $Sc\left({W}_{4}^{c};x\right)=96{x}^{1}+48{x}^{2}$ .

2) $\stackrel{\ast }{Sc}\left({W}_{6}^{c};x\right)=350{x}^{1}+295{x}^{2}+70{x}^{3}$ , $\stackrel{\ast }{Sc}\left({W}_{5}^{c};x\right)=260{x}^{1}+178{x}^{2}+8{x}^{3}$ , $\stackrel{\ast }{Sc}\left({W}_{4}^{c};x\right)=180{x}^{1}+60{x}^{2}$ .

2.4. Definition

A saw graph ${P}_{p}^{c}$ is a path of order p, say $\left\{{u}_{1},{u}_{2},\cdots ,{u}_{p}\right\}$ , with $p-1$ additional vertices $\left\{{v}_{1},{v}_{2},\cdots ,{v}_{p-1}\right\}$ and edges $\left\{{v}_{i}{u}_{i},{v}_{i}{u}_{i+1}:i=1,2,\cdots ,p-1\right\}$ as depicted in Figure 4.

It is clear $p\left({P}_{p}^{c}\right)=2p-1$ , $q\left({P}_{p}^{c}\right)=3\left(p-1\right)$ and $diam\text{\hspace{0.17em}}{P}_{p}^{c}=p-1$ , for $p\ge 2$ .

Theorem 2.4.1: For $p\ge 5$ , we have:

1) $Sc\left({P}_{p}^{c};x\right)=4\left(5p-7\right){x}^{1}+8\underset{k=2}{\overset{p-3}{\sum }}\left(3p-3k-1\right){x}^{k}+40{x}^{p-2}+16{x}^{p-1}.$

2) $\stackrel{*}{Sc}\left({P}_{p}^{c};x\right)=8\left(4p-7\right){x}^{1}+12\underset{k=2}{\overset{p-3}{\sum }}\left(3p-3k-2\right){x}^{k}+48{x}^{p-2}+16{x}^{p-1}.$

Proof:

For every vertice $y,z\in V\left({P}_{p}^{c}\right)$ , there is $d\left(y,z\right)=k$ , $1\le k\le p-1$ , and obviously $\underset{i=1}{\overset{p-1}{\sum }}|{D}_{i}|=\left(2p-1\right)\left(p-1\right)$ .

We will have four partitions for proof:

P1. if $d\left(y,z\right)=1$ , then $|{D}_{1}|=3\left(p-1\right)$ and is equal to $q\left({P}_{p}^{c}\right)$ , we have five subsets of it:

P1.1. $|\left\{\left({u}_{i},{u}_{i+1}\right):{u}_{i}{u}_{i+1}\in E\left({P}_{p}^{c}\right),{\delta }_{{u}_{i}}+{\delta }_{{u}_{i+1}}=6,{\delta }_{{u}_{i}}{\delta }_{{u}_{i+1}}=8,i=1,p-1\right\}|=2.$

P1.2. $\begin{array}{l}|\left\{\left({u}_{1},{v}_{1}\right),\left({u}_{p},{v}_{p-1}\right):{u}_{1}{v}_{1},{u}_{p}{v}_{p-1}\in E\left({P}_{p}^{c}\right),{\delta }_{{u}_{1}\left({u}_{p}\right)}+{\delta }_{{v}_{1}\left({v}_{p-1}\right)}=4,\\ {\delta }_{{u}_{1}\left({u}_{p}\right)}{\delta }_{{v}_{1}\left({v}_{p-1}\right)}=4\right\}|=2.\end{array}$

P1.3. $|\left\{\left({u}_{i},{v}_{i-1}\right):{u}_{i}{v}_{i-1}\in E\left({P}_{p}^{c}\right),{\delta }_{{u}_{i}}+{\delta }_{{v}_{i-1}}=6,{\delta }_{{u}_{i}}{\delta }_{{v}_{i-1}}=8,2\le i\le p-1\right\}|=p-2.$

P1.4. $|\left\{\left({u}_{i},{v}_{i}\right):{u}_{i}{v}_{i}\in E\left({P}_{p}^{c}\right),{\delta }_{{u}_{i}}+{\delta }_{{v}_{i}}=6,{\delta }_{{u}_{i}}{\delta }_{{v}_{i}}=8,2\le i\le p-1\right\}|=p-2.$

P1.5.

$|\left\{\left({u}_{i},{u}_{i+1}\right):{u}_{i}{u}_{i+1}\in E\left({P}_{p}^{c}\right),{\delta }_{{u}_{i}}+{\delta }_{{u}_{i+1}}=8,{\delta }_{{u}_{i}}{\delta }_{{u}_{i+1}}=16,2\le i\le p-2\right\}|=p-3.$

P2. if $d\left(y,z\right)=k$ , $2\le k\le p-3$ , then $\underset{k=2}{\overset{p-3}{\sum }}|{D}_{k}|=2\left(p-4\right)\left(p+1\right)$ , we have six subsets of it:

P2.1. $\begin{array}{l}|\left\{\left({u}_{1},{u}_{k+1}\right):{\delta }_{{u}_{1}}+{\delta }_{{u}_{k+1}}=6,{\delta }_{{u}_{1}}{\delta }_{{u}_{k+1}}=8\right\}\\ \cup \left\{\left({u}_{1},{v}_{k}\right):{\delta }_{{u}_{1}}+{\delta }_{{v}_{k}}=4,{\delta }_{{u}_{1}}{\delta }_{{v}_{k}}=4\right\}|=2.\end{array}$

P2.2. $\begin{array}{l}|\left\{\left({u}_{p},{u}_{p-k}\right):{\delta }_{{u}_{p}}+{\delta }_{{u}_{p-k}}=6,{\delta }_{{u}_{p}}{\delta }_{{u}_{p-k}}=8\right\}\\ \cup \left\{\left({u}_{p},{v}_{p-k}\right):{\delta }_{{u}_{p}}+{\delta }_{{v}_{p-k}}=4,{\delta }_{{u}_{p}}{\delta }_{{v}_{p-k}}=4\right\}|=2.\end{array}$

Figure 4. A saw graph ${P}_{p}^{c}$ .

P2.3. $\begin{array}{l}|\left\{\left({u}_{i},{u}_{k+i}\right):{\delta }_{{u}_{i}}+{\delta }_{{u}_{k+i}}=8,{\delta }_{{u}_{i}}{\delta }_{{u}_{k+i}}=16,2\le i\le p-k-1\right\}|\\ =p-k-2.\end{array}$

P2.4. $|\left\{\left({u}_{i},{v}_{k+i-1}\right):{\delta }_{{u}_{i}}+{\delta }_{{v}_{k+i-1}}=6,{\delta }_{{u}_{i}}{\delta }_{{v}_{k+i-1}}=8,2\le i\le p-k\right\}|=p-k-1.$

P2.5. $\begin{array}{l}|\left\{\left({u}_{p-i+1},{v}_{p-k-i+1}\right):{\delta }_{{u}_{p-i+1}}+{\delta }_{{v}_{p-k-i+1}}=6,{\delta }_{{u}_{p-i+1}}{\delta }_{{v}_{p-k-i+1}}=8,2\le i\le p-k\right\}|\\ =p-k-1.\end{array}$

P2.6. $|\left\{\left({v}_{i},{v}_{i+k-1}\right):{\delta }_{{v}_{i}}+{\delta }_{{v}_{i+k-1}}=4,{\delta }_{{v}_{i}}{\delta }_{{v}_{i+k-1}}=4,1\le i\le p-k\right\}|=p-k.$

P3. if $d\left(y,z\right)=p-2$ then $|{D}_{p-2}|=8$ , we have six subsets of it:

P3.1. $|\left\{\left({u}_{i},{u}_{p+i-2}\right):{\delta }_{{u}_{i}}+{\delta }_{{u}_{p+i-2}}=6,{\delta }_{{u}_{i}}{\delta }_{{u}_{p+i-2}}=8,i=1,2\right\}|=2.$

P3.2. $|\left\{\left({u}_{1},{v}_{p-2}\right):{\delta }_{{u}_{1}}+{\delta }_{{v}_{p-2}}=4,{\delta }_{{u}_{1}}{\delta }_{{v}_{p-2}}=4\right\}|=1.$

P3.3. $|\left\{\left({u}_{p},{v}_{2}\right):{\delta }_{{u}_{p}}+{\delta }_{{v}_{2}}=4,{\delta }_{{u}_{p}}{\delta }_{{v}_{2}}=4\right\}|=1.$

P3.4. $|\left\{\left({u}_{2},{v}_{p-1}\right):{\delta }_{{u}_{2}}+{\delta }_{{v}_{p-1}}=6,{\delta }_{{u}_{2}}{\delta }_{{v}_{p-1}}=8\right\}|=1.$

P3.5. $|\left\{\left({u}_{p-1},{v}_{1}\right):{\delta }_{{u}_{p-1}}+{\delta }_{{v}_{1}}=6,{\delta }_{{u}_{p-1}}{\delta }_{{v}_{1}}=8\right\}|=1.$

P3.6. $|\left\{\left({v}_{i},{v}_{p+i-3}\right):{\delta }_{{v}_{i}}+{\delta }_{{v}_{p+i-3}}=4,{\delta }_{{v}_{i}}{\delta }_{{v}_{p+i-3}}=4,i=1,2\right\}|=2.$

P4. if $d\left(y,z\right)=p-1$ then $|{D}_{p-1}|=4$ , we have four subsets of it:

P4.1. $|\left\{\left({u}_{1},{u}_{p}\right):{\delta }_{{u}_{1}}+{\delta }_{{u}_{p}}=4,{\delta }_{{u}_{1}}{\delta }_{{u}_{p}}=4\right\}|=1.$

P4.2. $|\left\{\left({u}_{1},{v}_{p-1}\right):{\delta }_{{u}_{1}}+{\delta }_{{v}_{p-1}}=4,{\delta }_{{u}_{1}}{\delta }_{{v}_{p-1}}=4\right\}|=1.$

P4.3. $|\left\{\left({u}_{p},{v}_{1}\right):{\delta }_{{u}_{p}}+{\delta }_{{v}_{1}}=4,{\delta }_{{u}_{p}}{\delta }_{{v}_{1}}=4\right\}|=1.$

P4.4. $|\left\{\left({v}_{1},{v}_{p-1}\right):{\delta }_{{v}_{1}}+{\delta }_{{v}_{p-1}}=4,{\delta }_{{v}_{1}}{\delta }_{{v}_{p-1}}=4\right\}|=1.$

From P1 - P4, we have:

$Sc\left({P}_{p}{}^{c};x\right)=4\left(5p-7\right)x+8\underset{k=2}{\overset{p-3}{\sum }}\left(3p-3k-1\right){x}^{k}+40{x}^{p-2}+16{x}^{p-1}.$

$\stackrel{*}{Sc}\left({P}_{p}^{c};x\right)=8\left(4p-7\right)x+12\underset{k=2}{\overset{p-3}{\sum }}\left(3p-3k-2\right){x}^{k}+48{x}^{p-2}+16{x}^{p-1}.$

Corollary2.4.2: For $p\ge 5$ , then:

1) $Sc\left({P}_{p}^{c}\right)=4\left(p+1\right){\left(p-1\right)}^{2}$ .

2) $\stackrel{*}{Sc}\left({P}_{p}^{c}\right)=6p{\left(p-1\right)}^{2}$ .

Corollary2.4.3: For $p\ge 5$ , then:

1) $10\frac{2}{3}\le \stackrel{¯}{Sc\left({P}_{p}^{c}\right)}\le 2p+1$ .

2) $13\frac{1}{3}\le \stackrel{¯}{\stackrel{*}{Sc}\left({P}_{p}^{c}\right)}<3\left(2p-1\right)/2$ .

Remark 2.4.4:

1) $Sc\left({P}_{3}^{c};x\right)=32{x}^{1}+16{x}^{2}$ , $Sc\left({P}_{4}^{c};x\right)=52{x}^{1}+40{x}^{2}+16{x}^{3}$ .

2) $\stackrel{*}{Sc}\left({P}_{3}^{c};x\right)=40{x}^{1}+16{x}^{2}$ , $\stackrel{*}{Sc}\left({P}_{4}^{c};x\right)=72{x}^{1}+48{x}^{2}+16{x}^{3}$ .

2.5. Definition

A Cog-Cycle is a graph ${C}_{p}^{c},p\ge 3$ obtained from a cycle graph ${C}_{p}=\left\{{u}_{1},{u}_{2},\cdots ,{u}_{p},{u}_{1}\right\}$ with p additional vertices $\left\{{v}_{1},{v}_{2},\cdots ,{v}_{p}\right\}$ , and edges $\left\{{v}_{i}{u}_{i},{v}_{i}{u}_{i+1}:i=1,2,\cdots ,p,\left({u}_{1}\equiv {u}_{p+1}\right)\right\}$ as shown in Figure 5.

It’s clear that $p\left({C}_{p}^{c}\right)=2p,q\left({C}_{p}^{c}\right)=3p$ , and

$diam\text{\hspace{0.17em}}{C}_{p}^{c}=\left[\begin{array}{l}\left(p/2\right)+1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}p\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{even}\text{\hspace{0.17em}}p\ge 4,\\ \left(p+1\right)/2,\text{\hspace{0.17em}}\text{\hspace{0.17em}}p\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{odd}\text{\hspace{0.17em}}p\ge 3.\end{array}$

Theorem 2.5.1: For $p\ge 6$ , then:

1) $Sc\left({C}_{p}^{C};x\right)=20px+24p\underset{k=2}{\overset{⌊\frac{p}{2}⌋-1}{\sum }}{x}^{k}+2p\left[\begin{array}{c}10{x}^{\frac{p}{2}}+{x}^{\frac{p}{2}+1},\text{​}\text{​}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}p\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{even},\\ 12{x}^{\frac{p-1}{2}}+5{x}^{\frac{p+1}{2}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}p\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{odd}.\end{array}$

2) $\stackrel{*}{Sc}\left({C}_{p}^{c};x\right)=32px+36p\underset{k=2}{\overset{⌊\frac{p}{2}⌋-1}{\sum }}{x}^{k}+2p\left[\begin{array}{c}14{x}^{\frac{p}{2}}+{x}^{\frac{p}{2}+1},\text{​}\text{​}\text{\hspace{0.17em}}\text{\hspace{0.17em}}p\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{even},\\ 18{x}^{\frac{p-1}{2}}+6{x}^{\frac{p+1}{2}},\text{​}\text{​}\text{\hspace{0.17em}}\text{\hspace{0.17em}}p\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{odd}.\end{array}$

Proof: For every vertice $y,z\in V\left({C}_{p}^{c}\right)$ , there is $d\left(y,z\right)=k$ , $1\le k\le ⌊\frac{p}{2}⌋+1$ , and obviously $\underset{i=1}{\overset{⌊\frac{p}{2}⌋+1}{\sum }}|{D}_{i}|=p\left(2p-1\right)$ . We will four partitions for proof:

P1. if $d\left(y,z\right)=1$ , then $|{D}_{i}|=3p$ and is equal to $q\left({C}_{p}^{c}\right)$ . We have three subsets of it:

P1.1.

$|\left({u}_{i},{u}_{i+1}\right):{u}_{i}{u}_{i+1}\in E\left({C}_{p}^{c}\right),{\delta }_{{u}_{i}}+{\delta }_{{u}_{i+1}}=8,{\delta }_{{u}_{i}}{\delta }_{{u}_{i+1}}=16,1\le i\le p,\left({u}_{p+1}\equiv {u}_{1}\right)|=p.$

P1.2. $|\left\{\left({u}_{i},{v}_{i}\right):{u}_{i}{v}_{i}\in E\left({C}_{p}^{c}\right),{\delta }_{{u}_{i}}+{\delta }_{{v}_{i}}=6,{\delta }_{{u}_{i}}{\delta }_{{v}_{i}}=8,1\le i\le p\right\}|=p.$

P1.3.

$|\left\{\left({u}_{i+1},{v}_{i}\right):{u}_{i+1}{v}_{i}\in E\left({C}_{p}^{c}\right),{\delta }_{{u}_{i+1}}+{\delta }_{{v}_{i}}=6,{\delta }_{{u}_{i+1}}{\delta }_{{v}_{i}}=8,1\le i\le p,\left({u}_{p+1}\equiv {u}_{1}\right)\right\}|=p$

P2. If $d\left(y,z\right)=k$ , $2\le k\le ⌊\frac{p}{2}⌋-1$ , then $\underset{k=2}{\overset{⌊\frac{p}{2}⌋-1}{\sum }}|{D}_{k}|=4p$ . We have four subsets of it:

Figure 5. A Cog-Cycle Graph ${C}_{p}^{c}$ .

when ${u}_{i}$ moving to ${v}_{j}$ clockwise.

P2.1. $|\left\{\left({u}_{i},{u}_{j}\right):{\delta }_{{u}_{i}}+{\delta }_{{u}_{j}}=8,{\delta }_{{u}_{i}}{\delta }_{{u}_{j}}=16,1\le i,j\le p,|i-j|=k,p-k\right\}|=p.$

P2.2. $|\left\{\left({u}_{i},{v}_{j}\right):{\delta }_{{u}_{i}}+{\delta }_{{v}_{j}}=6,{\delta }_{{u}_{i}}{\delta }_{{v}_{j}}=8,1\le i,j\le p,|i-j|=k-1,p-k+1\right\}|=p.$

when ${u}_{i}$ moving to ${v}_{j}$ reversed clockwise.

P2.3. $|\left\{\left({u}_{i},{v}_{j}\right):{\delta }_{{u}_{i}}+{\delta }_{{v}_{j}}=6,{\delta }_{{u}_{i}}{\delta }_{{v}_{j}}=8,1\le i,j\le p,|i-j|=k,p-k\right\}|=p.$

P2.4. $|\left\{\left({v}_{i},{v}_{j}\right):{\delta }_{{v}_{i}}+{\delta }_{{v}_{j}}=4,{\delta }_{{v}_{i}}{\delta }_{{v}_{j}}=4,1\le i,j\le p,|i-j|=k-1,p-k+1\right\}|=p.$

P3. If $d\left(y,z\right)=⌊p/2⌋$ , when p is even, then $|{D}_{p/2}|=7p/2$ , we have four subsets of it:

P3.1. $|\left\{\left({u}_{i},{u}_{i+p/2}\right):{\delta }_{{u}_{i}}+{\delta }_{{u}_{p/2}}=8,{\delta }_{{u}_{i}}{\delta }_{{u}_{p/2}}=16,1\le i\le p/2\right\}|=p/2.$

when ${u}_{i}$ moving to ${v}_{j}$ clockwise.

P3.2. $\begin{array}{l}|\left\{\left({u}_{i},{v}_{j}\right):{\delta }_{{u}_{i}}+{\delta }_{{v}_{j}}=6,{\delta }_{{u}_{i}}{\delta }_{{v}_{j}}=8,1\le i\le 1+\left(p/2\right)+1,j=\left(p/2\right)+i-1\right\}\\ \cup \left\{\left({u}_{i},{v}_{j}\right):{\delta }_{{u}_{i}}+{\delta }_{{v}_{j}}=6,{\delta }_{{u}_{i}}{\delta }_{{v}_{j}}=8,\left(p/2\right)+2\le i\le p,j=i-\left(p/2\right)-1\right\}|\\ =p.\end{array}$

when ${u}_{i}$ moving to ${v}_{j}$ reversed clockwise.

P3.3. $\begin{array}{l}|\left\{\left({u}_{i},{v}_{j}\right):{\delta }_{{u}_{i}}+{\delta }_{{v}_{j}}=6,{\delta }_{{u}_{i}}{\delta }_{{v}_{j}}=8,1\le i\le p/2,j=\left(p/2\right)+i\right\}\\ \cup \left\{\left({u}_{i},{v}_{j}\right):{\delta }_{{u}_{i}}+{\delta }_{{v}_{j}}=6,{\delta }_{{u}_{i}}{\delta }_{{v}_{j}}=8,1+\left(p/2\right)\le i\le p,j=i-\left(p/2\right)\right\}|\\ =p.\end{array}$

P3.4. $\begin{array}{l}|\left\{\left({v}_{i},{v}_{j}\right):{\delta }_{{v}_{i}}+{\delta }_{{v}_{j}}=4,{\delta }_{{v}_{i}}{\delta }_{{v}_{j}}=4,1\le i\le \left(p/2\right)+1,j=i+\left(p/2\right)-1\right\}\\ \cup \left\{\left({v}_{i},{v}_{j}\right):{\delta }_{{v}_{i}}+{\delta }_{{v}_{j}}=4,{\delta }_{{v}_{i}}{\delta }_{{v}_{j}}=4,2+\left(p/2\right)\le i\le p,j=i-\left(p/2\right)-1\right\}|\\ =p.\end{array}$

when p is odd, then $|{D}_{\left(p-1\right)/2}|=4p$ , we have four subsets of it:

P’3.1. $|\left\{\left({u}_{i},{u}_{j}\right):{\delta }_{{u}_{i}}+{\delta }_{{u}_{j}}=8,{\delta }_{{u}_{i}}{\delta }_{{u}_{j}}=16,1\le i,j\le p,|i-j|=⌊\frac{p}{2}⌋,⌈\frac{p}{2}⌉\right\}|=p.$

when ${u}_{i}$ moving to ${v}_{j}$ clockwise.

P’3.2. $\begin{array}{l}|\left\{\left({u}_{i},{v}_{j}\right):{\delta }_{{u}_{i}}+{\delta }_{{v}_{j}}=6,{\delta }_{{u}_{i}}{\delta }_{{v}_{j}}=8,1\le i\le \left(p+3\right)/2,j=i+\left(p-3\right)/2\right\}\\ \cup \left\{\left({u}_{i},{v}_{j}\right):{\delta }_{{u}_{i}}+{\delta }_{{v}_{j}}=6,{\delta }_{{u}_{i}}{\delta }_{{v}_{j}}=8,\left(p+5\right)/2\le i\le p,j=i-\left(p+3\right)/2\right\}|\\ =p.\end{array}$

when ${u}_{i}$ moving to ${v}_{j}$ reversed clockwise.

P’3.3. $\begin{array}{l}|\left\{\left({u}_{i},{v}_{j}\right):{\delta }_{{u}_{i}}+{\delta }_{{v}_{j}}=6,{\delta }_{{u}_{i}}{\delta }_{{v}_{j}}=8,1\le i\le \left(p-1\right)/2,j=i+\left(p+1\right)/2\right\}\\ \cup \left\{\left({u}_{i},{v}_{j}\right):{\delta }_{{u}_{i}}+{\delta }_{{v}_{j}}=6,{\delta }_{{u}_{i}}{\delta }_{{v}_{j}}=8,\left(p+1\right)/2\le i\le p,j=i-\left(p-1\right)/2\right\}|\\ =p.\end{array}$

P’3.4. $\begin{array}{l}|\left\{\left({v}_{i},{v}_{j}\right):{\delta }_{{v}_{i}}+{\delta }_{{v}_{j}}=4,{\delta }_{{v}_{i}}{\delta }_{{v}_{j}}=4,\text{\hspace{0.17em}}1\le i\le \left(p+3\right)/2,j=i+\left(p-3\right)/2\right\}\\ \cup \left\{\left({v}_{i},{v}_{j}\right):{\delta }_{{v}_{i}}+{\delta }_{{v}_{j}}=4,{\delta }_{{v}_{i}}{\delta }_{{v}_{j}}=4,\left(p+5\right)/2\le i\le p,j=i-\left(p-3\right)/2\right\}|\\ =p.\end{array}$

P4. If $d\left(y,z\right)=⌊p/2⌋+1$ , when p is even then $|{D}_{1+p/2}|=p/2$ , we have:

$|\left\{\left({v}_{i},{v}_{i+p/2}\right):{\delta }_{{v}_{i}}+{\delta }_{{v}_{i+p/2}}=4,{\delta }_{{v}_{i}}{\delta }_{{v}_{i+p/2}}=4,1\le i\le p/2\right\}|=p/2.$

when p is odd then $|{D}_{\left(p+1\right)/2}|=2p$ , we have two subsets of it:

P4.1. $\begin{array}{l}|\left\{\left({u}_{i},{v}_{j}\right):{\delta }_{{u}_{i}}+{\delta }_{{v}_{j}}=6,{\delta }_{{u}_{i}}{\delta }_{{v}_{j}}=8,1\le i\le \left(p+1\right)/2,j=i+\left(p-1\right)/2\right\}\\ \cup \left\{\left({u}_{i},{v}_{j}\right):{\delta }_{{u}_{i}}+{\delta }_{{v}_{j}}=6,{\delta }_{{u}_{i}}{\delta }_{{v}_{j}}=8,\left(p+3\right)/2\le i\le p,j=i-\left(p+1\right)/2\right\}|\\ =p.\end{array}$

P4.2. $\begin{array}{l}|\left\{\left({v}_{i},{v}_{j}\right):{\delta }_{{v}_{i}}+{\delta }_{{v}_{j}}=4,{\delta }_{{v}_{i}}{\delta }_{{v}_{j}}=4,1\le i\le \left(p+1\right)/2,j=i+\left(p-1\right)/2\right\}\\ \cup \left\{\left({v}_{i},{v}_{j}\right):{\delta }_{{v}_{i}}+{\delta }_{{v}_{j}}=4,{\delta }_{{v}_{i}}{\delta }_{{v}_{j}}=4,\left(p+3\right)/2\le i\le p,j=i-\left(p+1\right)/2\right\}|\\ =p.\end{array}$

From P1 - P4, we have:

$Sc\left({C}_{p}^{c};x\right)=20px+24p\underset{k=2}{\overset{⌊p/2⌋-1}{\sum }}{x}^{k}+2p\left[\begin{array}{l}10{x}^{\frac{p}{2}}+{x}^{\frac{p}{2}+1};\text{\hspace{0.17em}}\text{\hspace{0.17em}}p\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{even},\\ 12{x}^{\frac{p-1}{2}}+5{x}^{\frac{p+1}{2}};\text{\hspace{0.17em}}\text{\hspace{0.17em}}p\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{odd}.\end{array}$

$\stackrel{*}{Sc}\left({C}_{p}^{c};x\right)=32px+36p\underset{k=2}{\overset{⌊p/2⌋-1}{\sum }}{x}^{k}+2p\left[\begin{array}{c}14{x}^{\frac{p}{2}}+{x}^{\frac{p}{2}+1};\text{\hspace{0.17em}}\text{\hspace{0.17em}}p\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{even},\\ 18{x}^{\frac{p-1}{2}}+6{x}^{\frac{p+1}{2}};\text{\hspace{0.17em}}\text{\hspace{0.17em}}p\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{odd}.\end{array}$

Corollary2.5.2: For $p\ge 6$ , then:

1) $Sc\left({C}_{p}^{c}\right)=p\left(3{p}^{2}+5p-2\right)$ .

2) $\stackrel{*}{Sc}\left({C}_{p}^{c}\right)=\frac{p}{2}\left[\begin{array}{l}9{p}^{2}+12p-4;\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}p\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{even},\\ 9{p}^{2}+12p-5;\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}p\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{odd}.\end{array}$

Corollary2.5.3: For $p\ge 6$ , then:

1) $12\frac{4}{11}\le \stackrel{¯}{Sc}\left({C}_{p}^{c}\right)<\left(3p+7\right)/2$ .

2) $17.7<\stackrel{¯}{\stackrel{*}{Sc}}\left({C}_{p}^{c}\right)<\left(18p+53\right)/8$ .

Remark 2.5.4:

1) $Sc\left({C}_{4}^{c};x\right)=80{x}^{1}+80{x}^{2}+8{x}^{3}$ , $Sc\left({C}_{5}^{c};x\right)=100{x}^{1}+120{x}^{2}+50{x}^{3}$ .

2) $\stackrel{*}{Sc}\left({C}_{4}^{c};x\right)=128{x}^{1}+112{x}^{2}+8{x}^{3}$ , $\stackrel{*}{Sc}\left({C}_{5}^{c};x\right)=160{x}^{1}+180{x}^{2}+60{x}^{3}$ .

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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