The Eccentric-Distance Sum Polynomials of Graphs by Using Graph Products

Abstract

The correlations between the physico-chemical properties of a chemical structure and its molecular structure-properties are used in quantitative structure-activity and property relationship studies (QSAR/QSPR) by using graph-theoretical analysis and techniques. It is well known that some structure-activity and quantitative structure-property studies, using eccentric distance sum, are better than the corresponding values obtained by using the Wiener index. In this article, we give precise expressions for the eccentric distance sum polynomial of some graph products such as join, Cartesian, lexicographic, corona and generalized hierarchical products. We implement our outcomes to calculate this polynomial for some significant families of molecular graphs in the form of the above graph products.

1. Introduction

Let $\mathcal{H}$ be an n-vertex simple and a connected graph with size e. We represent the edge set and the vertex set of $\mathcal{H}$ by $\mathcal{E}\left(\mathcal{H}\right)$ and $\mathcal{V}\left(\mathcal{H}\right)$, respectively. The degree of $h\in \mathcal{V}\left(\mathcal{H}\right)$ is the number of adjacent vertices to h in $\mathcal{H}$, and it is represented by $de{g}_{\mathcal{H}}\left(h\right)$. For $h\in \mathcal{V}\left(\mathcal{H}\right)$, ${\gamma }_{\mathcal{H}}\left(h\right)$ is the set of all neighbors of h in $\mathcal{H}$. For the two given vertices, $h_p,h_q\in \mathcal{V}\left(\mathcal{H}\right)$, we describe their distance $d_{\mathcal{H}}(h_p,h_q)$ by the length of the shortest path among $h_p$ and $h_q$ in $\mathcal{H}$, and $D_{\mathcal{H}}\left(h_p\right)={\sum }_{h_q\in \mathcal{V}\left(\mathcal{H}\right)}{d}_{\mathcal{H}}(h_p,h_q)$. For a vertex $h_p\in \mathcal{V}\left(\mathcal{H}\right)$, its eccentricity $e{c}_{\mathcal{H}}\left(h_p\right)$ is written as $e{c}_{\mathcal{H}}\left(h_p\right)=\underset{h_q\in \mathcal{V}\left(\mathcal{H}\right)}{max}{d}_{\mathcal{H}}(h_p,h_q)$. A vertex h is named as a well-connected node in an n-vertex graph $\mathcal{H}$ if $de{g}_{\mathcal{H}}\left(h\right)=n-1$ and obviously its eccentricity will be 1. We represent the set of all well-connected vertices of $\mathcal{H}$ with $\mathcal{N}\left(\mathcal{H}\right)$, and $\left|\mathcal{N}\right(\mathcal{H}\left)\right|$ represents its cardinality. The unique vertex of $K_1$ is considered to be a well-connected vertex, and $K_1$ is a well-connected graph. Let $K_{a_1,a_2,\cdots ,a_l}$ be a complete l-partite graph with an ${\displaystyle \sum _{k=1}^l}{a}_k$ number of vertices. The set of vertices $\mathcal{V}\left(K_{a_1,a_2,\cdots ,a_l}\right)$ can be divided up into subsets ${\mathcal{V}}_1,{\mathcal{V}}_2,\cdots ,{\mathcal{V}}_l$ such that there is no edge between the vertices of the same set ${\mathcal{V}}_k$, $1\le k\le l$. We use the notions $K_n$ and $C_n$ for the n-vertex complete graph and cycle, respectively.

In mathematical chemistry and chemical graph theory, a molecular (or chemical) graph is a representation of the structural formula of a chemical compound in the form of graph theory. Its vertices are presented by the types of the corresponding atoms, and its edges are presented by the kinds of its bonds. A molecular graph has been used in the prediction of properties relevant to the drug discovery and optimization process. For example, knowledge discovery can be used for the identification of lead compounds in pharmaceutical data matching. A molecular descriptor is a numerical parameter of a graph that distinguishes its topology. Topological invariants have constructed numerous applications in pharmaceutical drug design, QSAR/QSPR study, chemical documentations and isomer discrimination. The eccentric-connectivity index of $\mathcal{H}$ is a new distance-related descriptor that was designed by Sharma et al. [1] and is described as:

$${\xi }^c\left(\mathcal{H}\right)=\sum _{h\in \mathcal{V}\left(\mathcal{H}\right)}de{g}_{\mathcal{H}}\left(h\right)e{c}_{\mathcal{H}}\left(h\right).$$

In the present-day literature, a modification of ${\xi }^c\left(\mathcal{H}\right)$ has been found and mentioned as the total-eccentricity $\tau \left(\mathcal{H}\right)$ index.It is described as:

$$\tau \left(\mathcal{H}\right)=\sum _{h\in \mathcal{V}\left(\mathcal{H}\right)}e{c}_{\mathcal{H}}\left(h\right).$$

The eccentric connectivity polynomial ${\xi }^c(\mathcal{H},y)$ is the continuation of ${\xi }^c$, which was initiated by Ashraf and Jalali [2], and described as:

$$\begin{array}{ccc}\hfill {\xi }^c(\mathcal{H},y)& =& \sum _{h\in \mathcal{V}\left(\mathcal{H}\right)}de{g}_{\mathcal{H}}\left(h\right)y^{e{c}_{\mathcal{H}}\left(h\right)}.\hfill \end{array}$$

The total eccentricity polynomial of $\mathcal{H}$ is interpreted as follows:

$$\tau (\mathcal{H},y)=\sum _{h\in \mathcal{V}\left(\mathcal{H}\right)}{y}^{e{c}_{\mathcal{H}}\left(h\right)}.$$

It is straightforward that the total-eccentricity and eccentric-connectivity indices of $\mathcal{H}$ can be acquired from the parallel polynomials by taking their first derivative at $y=1$.

In theoretical chemistry, the research of an eccentric-connectivity polynomial for chemical structures has become an attractive area in recent years. The eccentric distance sum of $\mathcal{H}$ was proposed by Gupta et al. [3] and was construed as follows:

$${\xi }^{ds}\left(\mathcal{H}\right)=\sum _{\{h_p,h_q\}\subseteq \mathcal{V}\left(\mathcal{H}\right)}(e{c}_{\mathcal{H}}\left(h_p\right)+e{c}_{\mathcal{H}}\left(h_q\right))d_{\mathcal{H}}(h_p,h_q)=\sum _{h_p\in \mathcal{V}\left(\mathcal{H}\right)}e{c}_{\mathcal{H}}\left(h_p\right)D_{\mathcal{H}}\left(h_p\right).$$

This index suggests a massive capability for structure-activity and structure-property connections and has exhibited a great predictive power in pharmaceutical properties. It also shows vast discriminating power with reference to physical and biological characterizations [3]. From [3], the eccentric-distance sum gives a better result in the study of some structure-activities and quantitative structure-properties, more than the corresponding results acquired by using the Wiener index. WIth the help of the eccentric-distance sum index and anti-HIV activity of dihydroseselins, the researchers investigated and facilitated the development of potent and safe anti-HIV agents. The accuracy of prediction was found to be more than 88 percent with regard to anti-HIV activity. The eccentric distance sum exhibited a much better correlation and fewer average errors than Wiener’s topological index. Recently, the mathematical aspects of the ${\xi }^{ds}$ have been examined. The eccentric-distance sum polynomial of $\mathcal{H}$ is proposed as:

$${\xi }^{ds}(\mathcal{H},y)=\sum _{h_p\in \mathcal{V}\left(\mathcal{H}\right)}{D}_{\mathcal{H}}\left(h_p\right)y^{e{c}_{\mathcal{H}}\left(h_q\right)}=\sum _{h_p\in \mathcal{V}\left(\mathcal{H}\right)}\sum _{h_q\in \mathcal{V}\left(\mathcal{H}\right)}{d}_{\mathcal{H}}(h_p,h_q)y^{e{c}_{\mathcal{H}}\left(h_p\right)}.$$

(1)

Graph operations play a valuable part in many combinatorial characterizations, graph decompositions into isomorphic subgraphs in pure mathematics, and also applied mathematics. The study of finding the topological invariants of different molecular graph structures by the use of the topological invariants of graph operations has valuable importance these days. There are many complicated molecular structures, such as $C_4$-nanotorus, Hamming graphs, closed fence graphs, etc., for which it is so difficult to find their topological indices and polynomials, so for this purpose, the researchers can use graph operations. Azari and Iranmanesh [4] determined this index for some prominent graph operations. Došlić and Hosseinzadeh [5] computed its corresponding polynomial for various graph operations. Yu et al. [6] worked out this index for unicyclic graphs and trees. Hua et al. [7] secured the bound on the eccentric-distance sum of cacti. Hua et al. [8] considered the graphs having the smallest eccentric-distance sum with graph parameters. Ilić et al. [9] computed several bounds of ${\xi }^{ds}$ in the form of the different graph invariants. Geng et al. [10] worked out the ${\xi }^{ds}$ of the trees with some given parameters. For more information on different aspects of indices and polynomials, one can see [11,12,13,14,15,16,17,18,19,20].

In this article, we are interested in acquiring the eccentric-distance sum polynomial of graph products. Then, we perform our findings to determine the eccentric-distance sum polynomial of certain significant categories of graphs in the form of factors of graph products.

Proposition 1.

Let $K_n$ and $C_n$ be the n-vertex complete graph and cycle, respectively. Then

1 

${\xi }^{ds}(K_n,y)=n(n-1)y$.

2 

$\tau (K_n,y)=ny$.

3 

${\xi }^{ds}(C_n,y)=\left\{\begin{array}{cc}{\displaystyle \frac{n^3}{4}}{y}^{(n/2)},\hfill & forniseven,\hfill \\ {\displaystyle \frac{n(n^2-1)}{4}}{y}^{\left(\right(n-1)/2)},\hfill & fornisodd.\hfill \end{array}\right.$

4 

$\tau (C_n,y)=\left\{\begin{array}{cc}n{y}^{(n/2)},\hfill & forniseven,\hfill \\ n{y}^{\left(\right(n-1)/2)},\hfill & fornisodd.\hfill \end{array}\right.$

2. Join

The join of ${\mathcal{H}}_p$ and ${\mathcal{H}}_q$ graphs is recognized as ${\mathcal{H}}_p+{\mathcal{H}}_q$, which consists of edge sets $\mathcal{E}\left({\mathcal{H}}_l\right)$ and disjoint vertex sets $\mathcal{V}\left({\mathcal{H}}_l\right)$, where $l=p,q$. It is graph union ${\mathcal{H}}_p\cup {\mathcal{H}}_q$ including all the links joining $\mathcal{V}\left({\mathcal{H}}_p\right)$ and $\mathcal{V}\left({\mathcal{H}}_q\right)$. For $\mathcal{H}=\underset{ktimes}{\underbrace{{\mathcal{H}}_1+{\mathcal{H}}_2+\cdots +{\mathcal{H}}_k}}$, we represent $\mathcal{H}$ by $k\mathcal{H}$. First, we give a lemma, which plays a vital role in the solution of the leading result.

Lemma 1.

Let ${\mathcal{H}}_1,{\mathcal{H}}_2,\cdots ,{\mathcal{H}}_k$ be graphs with $|\mathcal{V}\left({\mathcal{H}}_l\right)|=n_l$ and $|\mathcal{E}\left({\mathcal{H}}_l\right)|=e_l$, $1\le l\le k$. Then, for the vertices $h_p,g_p\in \mathcal{V}({\mathcal{H}}_1+\cdots +{\mathcal{H}}_k)$, we have the following:

1 

$d_{{\mathcal{H}}_1+{\mathcal{H}}_2+\cdots +{\mathcal{H}}_k}(h_p,g_p)=\left\{\begin{array}{cc}0,\hfill & \mathit{if}{h}_p=g_p,\hfill \\ 1,\hfill & \begin{array}{cc}\hfill \mathit{if}{g}_p{h}_p\in \mathcal{E}\left({\mathcal{H}}_l\right)& or{h}_p\in \mathcal{V}\left({\mathcal{H}}_l\right),g_p\in \mathcal{V}\left({\mathcal{H}}_j\right),\hfill \\ & 1\le l\ne j\le k,\hfill \end{array}\hfill \\ 2,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$

2 

$e{c}_{{\mathcal{H}}_1+{\mathcal{H}}_2+\cdots +{\mathcal{H}}_k}\left(h_p\right)=\left\{\begin{array}{cc}1,\hfill & if h_p\in \mathcal{N}\left({\mathcal{H}}_l\right),1\le l\le k,\hfill \\ 2,\hfill & otherwise.\hfill \end{array}\right.$

Next, we calculate the eccentric distance sum polynomial of ${\mathcal{H}}_1+{\mathcal{H}}_2+\cdots +{\mathcal{H}}_k$ with the help of Lemma 1.

Theorem 1.

For a graph $\mathcal{H}={\mathcal{H}}_1+{\mathcal{H}}_2+\cdots +{\mathcal{H}}_k$, where $|\mathcal{V}\left({\mathcal{H}}_m\right)|=n_m$, $|\mathcal{E}\left({\mathcal{H}}_m\right)|=e_m$, and $1\le m\le k$, we have

$$\begin{array}{ccc}\hfill {\xi }^{ds}(\mathcal{H},y)& =& \sum _{m=1}^k|\mathcal{N}\left({\mathcal{H}}_m\right)|\left(\sum _{m=1}^k{n}_m-1\right)y+\sum _{m=1}^k(n_m^2-2n_m-2e_m+\left|\mathcal{N}\left({\mathcal{H}}_m\right)\right|+n_m\sum _{j=1}^k{n}_j\hfill \\ & -& |\mathcal{N}\left({\mathcal{H}}_m\right)|\sum _{j=1}^k{n}_j)y^2.\hfill \end{array}$$

Proof. 

Note that if $h\in \mathcal{N}\left({\mathcal{H}}_m\right)$, then $de{g}_{{\mathcal{H}}_m}\left(u\right)=n_m-1$, for $1\le m\le k$. The degree of $h\in \mathcal{V}\left(\mathcal{H}\right)$ is $de{g}_{\mathcal{H}}\left(h\right)=de{g}_{{\mathcal{H}}_m}\left(h\right)+{\displaystyle {\displaystyle \sum }_{j=1,j\ne m}^k}{n}_j$, where $h\in \mathcal{V}\left({\mathcal{H}}_i\right)$ and $1\le m\le k$. Now, by Equation (1) and Lemma 1, we have:

$$\begin{array}{ccc}\hfill {\xi }^{ds}(\mathcal{H},y)& =& \sum _{m=1}^k[\sum _{h\in \mathcal{N}\left({\mathcal{H}}_m\right)}\sum _{g\in \mathcal{V}\left(\mathcal{H}\right)\backslash \left\{h\right\}}\left(1\right)y^{\left(1\right)}+\sum _{h\in \mathcal{V}\left({\mathcal{H}}_m\right)}\sum _{h\in {\gamma }_{{\mathcal{H}}_m}\left(h\right)\cup {\displaystyle \bigcup _{m\ne j=1}^k}\mathcal{V}\left({\mathcal{H}}_j\right)}\left(1\right)y^{\left(2\right)}\hfill \\ & +& \sum _{h\in \mathcal{V}\left({\mathcal{H}}_m\right)\backslash \mathcal{N}\left({\mathcal{H}}_m\right)}\sum _{g\in \mathcal{V}\left({\mathcal{H}}_m\right)\backslash ({\gamma }_{{\mathcal{H}}_m}\left(h\right)\cup \left\{h\right\})}\left(2\right)y^{\left(2\right)}\left]\right]\hfill \\ & =& \sum _{m=1}^k[|\mathcal{N}\left({\mathcal{H}}_m\right)|\left(\sum _{m=1}^k{n}_m-1\right)y+y^2\sum _{h\in \mathcal{V}\left({\mathcal{H}}_m\right)\backslash \mathcal{N}\left({\mathcal{H}}_m\right)}\left(de{g}_{{\mathcal{H}}_m}\left(h\right)+\sum _{m\ne j=1}^k{n}_j\right)\hfill \\ & +& 2y^2\sum _{h\in \mathcal{V}({\mathcal{H}}_m)\backslash \mathcal{N}({\mathcal{H}}_m)}(n_m-1-de{g}_{{\mathcal{H}}_m}\left(h\right))]\hfill \\ & =& \sum _{m=1}^k\left[\right|\mathcal{N}\left({\mathcal{H}}_m\right)|\left(\sum _{m=1}^k{n}_m-1\right)y+y^2(2e_m-(n_m-1)|\mathcal{N}\left({\mathcal{H}}_m\right)|+(n_m\hfill \\ & -& |\mathcal{N}\left({\mathcal{H}}_m\right)\left|\right)\sum _{j=1,j\ne m}^k{n}_j)+2y^2((n_m-1)(n_m-|\mathcal{N}\left({\mathcal{H}}_m\right)\left|\right)-(2e_m\hfill \\ & -& (n_m-1)|\mathcal{N}\left({\mathcal{H}}_m\right)\left|\right)\left)\right].\hfill \end{array}$$

After some simple computations, we can get above the expression. This finishes the proof. □

Similarly, the following corollaries can be easily derived from Theorem 1, so we omitted their proofs.

Corollary 1.

For graphs ${\mathcal{H}}_p$ and ${\mathcal{H}}_q$ with $|\mathcal{V}\left({\mathcal{H}}_m\right)|=n_m$ and $|\mathcal{E}\left({\mathcal{H}}_m\right)|=e_m$, where $m=p$ or q, we have

$$\begin{array}{ccc}\hfill {\xi }^{ds}({\mathcal{H}}_p+{\mathcal{H}}_q,y)& =& 2(n_1^2+n_2^2+n_1{n}_2-n_1-n_2-e_1-e_2)y^2-(n_1+n_2-1)\left(\right|\mathcal{N}\left({\mathcal{H}}_p\right)|\hfill \\ & +& |\mathcal{N}\left({\mathcal{H}}_q\right)\left|\right)y(1-y).\hfill \end{array}$$

Corollary 2.

Let ${\mathcal{H}}_1={\mathcal{H}}_2=\cdots ={\mathcal{H}}_k=\mathcal{H}$, and let $k\mathcal{H}$ denote the k copies of $\mathcal{H}$ with order n and size e. Then,

$${\xi }^{ds}(k\mathcal{H},y)=k(1-kn)\left|\mathcal{N}\left(\mathcal{H}\right)\right|(y^2-y)+\left((k+1)n^2-2n-2e\right)k{y}^2.$$

Example 1.

It is easy to observe that, for the given complete m-partite graph $K_{a_1,a_2,\cdots ,a_m}={\overline{K}}_{a_1}+{\overline{K}}_{a_2}+\cdots +{\overline{K}}_{a_m}$, using Theorem 1, we acquire

$${\xi }^{ds}(K_{a_1,a_2,\cdots ,a_m},y)=\left(\sum _{p=1}^m{a}_p^2-2\sum _{p=1}^m{a}_p+{\left(\sum _{p=1}^m{a}_p\right)}^2\right)y^2.$$

For $m=2$, we have complete bipartite graph $K_{a_1,a_2}$ on $a_1+a_2$ vertices. Therefore, ${\xi }^{ds}(K_{a_1,a_2},y)$$=2(a_1^2+a_2^2+a_1{a}_2-a_1-a_2)$.

Example 2.

The cone graph $C_{m,n}$ is described as $C_m+{\overline{K}}_n$, and its eccentric-distance sum polynomial is ${\xi }^{ds}(C_{m,n},y)=2(m^2+n^2+mn-2m-n)y^2$.

For a given graph $\mathcal{H}$, the suspension of $\mathcal{H}$ is defined as $K_1+\mathcal{H}$. The following result follows from Corollary 1.

Example 3.

For a graph $\mathcal{H}$ with $\left|\mathcal{E}\right(\mathcal{H}\left)\right|=e$ and $\left|\mathcal{V}\right(\mathcal{H}\left)\right|=n$, we have

$${\xi }^c(K_1+\mathcal{H},y)=2(e-n)y^2-n(1+|\mathcal{N}\left(\mathcal{H}\right)\left|\right)(y-1)y.$$

(2)

Star graph $S_{n+1}$, fan graph $F_{n+1}$ and wheel graph $W_{n+1}$ on $n+1$ vertices are the suspensions of the empty graph, $\overline{K_n}$, $P_n$ and $C_n$, respectively (see Figure 1). Therefore, by (2), we obtain the following:

$$\begin{array}{cc}\hfill \begin{array}{c}\hfill {\xi }^{ds}(S_{n+1},y)=ny+n(2n-1)y^2,\end{array}& {\xi }^{ds}(F_{n+1},y)=ny+2y^2+n(2n-3)y^2,\\ \hfill {\xi }^{ds}(W_{n+1},y)=ny+n(2n-3)y^2.\end{array}$$

3. Cartesian Product

Here, we denote the Cartesian product of ${\mathcal{H}}_p$ and ${\mathcal{H}}_q$ by ${\mathcal{H}}_p\otimes {\mathcal{H}}_q.$ It has $\mathcal{V}\left({\mathcal{H}}_p\right)\times \mathcal{V}\left({\mathcal{H}}_q\right)$ vertex set and $(h_p,g_p)(h_q,g_q)\in \mathcal{E}({\mathcal{H}}_p\otimes {\mathcal{H}}_q)$ if $h_p=h_q$ and $g_p{g}_q\in \mathcal{E}\left({\mathcal{H}}_q\right)$, or $h_p{h}_q\in \mathcal{E}\left({\mathcal{H}}_p\right)$ and $g_p=g_q$. In the beginning, we give an essential lemma that relates the properties of the Cartesian product.

Lemma 2.

For ${\mathcal{H}}_p$ and ${\mathcal{H}}_q$ with $|\mathcal{V}\left({\mathcal{H}}_l\right)|=n_l$ and $|\mathcal{E}\left({\mathcal{H}}_l\right)|=e_l$, where $l=p$ or q, we have the following:

(a) 

$d_{{\mathcal{H}}_p\otimes {\mathcal{H}}_q}((h_p,g_p),(h_q,g_q))=d_{{\mathcal{H}}_p}(h_p,h_q)+d_{{\mathcal{H}}_q}(g_p,g_q).$

(b) 

$e{c}_{{\mathcal{H}}_p\otimes {\mathcal{H}}_q}\left((h_p,g_p)\right)=e{c}_{{\mathcal{H}}_p}\left(h_p\right)+e{c}_{{\mathcal{H}}_q}\left(g_p\right).$

Now, we give the expression for the eccentric-distance sum polynomial of ${\mathcal{H}}_1\otimes {\mathcal{H}}_2\otimes \cdots \otimes {\mathcal{H}}_k$, in the form of eccentric-distance sum, and total eccentricity polynomials of ${\mathcal{H}}_m$, for $1\le m\le k$.

Theorem 2.

Let ${\mathcal{H}}_1,{\mathcal{H}}_2,\cdots ,{\mathcal{H}}_k$ be graphs with $|\mathcal{V}\left({\mathcal{H}}_m\right)|=n_m$, $|\mathcal{E}\left({\mathcal{H}}_m\right)|=e_m$, $1\le m\le k$, and $n={\displaystyle \prod _{m=1}^k}{n}_m$. Then we have

$${\xi }^{ds}({\mathcal{H}}_1\otimes {\mathcal{H}}_2\otimes \cdots \otimes {\mathcal{H}}_k,y)=\sum _{m=1}^k{\xi }^{ds}({\mathcal{H}}_m,y)\prod _{r=1,r\ne m}^k{n}_r\tau ({\mathcal{H}}_r,y).$$

Proof. 

By using Lemma 2 and Equation (1), we obtain

$$\begin{array}{ccc}\hfill {\xi }^{ds}({\mathcal{H}}_p\otimes {\mathcal{H}}_q,y)& =& \sum _{(h_p,g_p)\in \mathcal{V}({\mathcal{H}}_p\otimes {\mathcal{H}}_q)}\sum _{(h_q,g_q)\in \mathcal{V}({\mathcal{H}}_p\otimes {\mathcal{H}}_q)}{d}_{{\mathcal{H}}_p\otimes {\mathcal{H}}_q}((h_p,g_p),(h_q,g_q))y^{e{c}_{{\mathcal{H}}_p\otimes {\mathcal{H}}_q}\left((h_p,g_p)\right)}\hfill \\ & =& \sum _{h_p\in \mathcal{V}\left({\mathcal{H}}_p\right)}\sum _{g_p\in \mathcal{V}\left({\mathcal{H}}_q\right)}\sum _{h_q\in \mathcal{V}\left({\mathcal{H}}_p\right)}\sum _{g_q\in \mathcal{V}\left({\mathcal{H}}_q\right)}(d_{{\mathcal{H}}_p}(h_p,h_q)+d_{{\mathcal{H}}_q}(g_p,g_q))\hfill \\ & & y^{(e{c}_{{\mathcal{H}}_p}\left(h_p\right)+e{c}_{{\mathcal{H}}_q}\left(g_p\right))}\hfill \\ & =& \sum _{h_p\in \mathcal{V}\left({\mathcal{H}}_p\right)}\sum _{g_p\in \mathcal{V}\left({\mathcal{H}}_p\right)}{d}_{{\mathcal{H}}_p}(h_p,g_p)y^{e{c}_{{\mathcal{H}}_p}\left(h_p\right)}\sum _{h_q\in \mathcal{V}\left({\mathcal{H}}_q\right)}\sum _{g_q\in \mathcal{V}\left({\mathcal{H}}_q\right)}{y}^{e{c}_{{\mathcal{H}}_q}\left(h_q\right)}\hfill \\ & +& \sum _{h_p\in \mathcal{V}\left({\mathcal{H}}_p\right)}\sum _{g_p\in \mathcal{V}\left({\mathcal{H}}_p\right)}{y}^{e{c}_{{\mathcal{H}}_p}\left(h_p\right)}\sum _{h_q\in \mathcal{V}\left({\mathcal{H}}_q\right)}\sum _{g_q\in \mathcal{V}\left({\mathcal{H}}_q\right)}{d}_{{\mathcal{H}}_q}(h_q,g_q)y^{e{c}_{{\mathcal{H}}_q}\left(h_q\right)}\hfill \\ & =& n_2{\xi }^{ds}({\mathcal{H}}_p,y)\tau ({\mathcal{H}}_q,y)+n_1{\xi }^{ds}({\mathcal{H}}_q,y)\tau ({\mathcal{H}}_p,y).\hfill \end{array}$$

Again, by Lemma 2, we have:

$$\begin{array}{ccc}\hfill {\xi }^{ds}\left(\underset{m=1}{\overset{k}{⨂}}{\mathcal{H}}_m,y\right)& =& {\xi }^{ds}\left(\left(\underset{m=1}{\overset{k-1}{⨂}}{\mathcal{H}}_m\right)⨂{\mathcal{H}}_k,y\right)\hfill \\ & =& {\xi }^{ds}\left(\underset{m=1}{\overset{k-1}{⨂}}{\mathcal{H}}_m,y\right)n_k\tau ({\mathcal{H}}_k,y)+\prod _{r=1}^{k-1}{n}_r{\xi }^{ds}({\mathcal{H}}_k,y)\tau \left(\underset{m=1}{\overset{k-1}{⨂}}{\mathcal{H}}_m,y\right)\hfill \\ & =& \left(\sum _{m=1}^{k-1}{\xi }^{ds}({\mathcal{H}}_m,y)\prod _{r=1,r\ne m}^{k-1}{n}_r\tau ({\mathcal{H}}_r,y)\right)n_k\tau ({\mathcal{H}}_k,y)\hfill \\ & +& \prod _{r=1}^{k-1}{n}_r\tau ({\mathcal{H}}_r,y){\xi }^{ds}({\mathcal{H}}_k,y)\hfill \\ & =& \sum _{m=1}^k{\xi }^{ds}({\mathcal{H}}_m,y)\prod _{r=1,r\ne i}^k{n}_r\tau ({\mathcal{H}}_r,y).\hfill \end{array}$$

This finishes the proof. □

Example 4.

Let $S=C_a\otimes C_b$, for some integers $a,b\ge 3$, denote a $C_4$-nanotorus. Then, by using Theorem 2 and Proposition 1, we obtain

$$\begin{array}{ccc}\hfill {\xi }^{ds}(S,y)& =& \left\{\begin{array}{cc}{\displaystyle \frac{ab(b^3+a^3)}{4}}{y}^{(b+a)/2},\hfill & if b and a are even,\hfill \\ {\displaystyle \frac{ab(b^3+a^3-a)}{4}}{y}^{(b+a-1)/2},\hfill & if b is even and a is odd,\hfill \\ {\displaystyle \frac{ab(b^3+a^3-b)}{4}}{y}^{(b+a-1)/2},\hfill & if a is even and b is odd,\hfill \\ {\displaystyle \frac{ab(b^3+a^3-b-a)}{4}}{y}^{(b+a-2)/2},\hfill & if b and a are odd.\hfill \end{array}\right.\hfill \end{array}$$

Example 5.

The Hamming graph is a connected graph with k-tuples vertices $h_1{h}_2\cdots h_k$, where $h_l\in \{0,1,\cdots ,n_l-1\}$, $n_l\ge 2$, $1\le l\le k$, and $(h_p,h_q,\cdots ,h_k)(g_1,g_2,\cdots ,g_k)$ is an edge, if the interrelated tuples differ in exactly one place and are commonly expressed as $H_{n_1,n_2,\cdots ,n_k}={\displaystyle \underset{l=1}{\overset{k}{⨂}}}{K}_{n_l}$. By using Theorem 2 and Proposition 1, we obtain:

$${\xi }^{ds}(H_{n_1,n_2,\cdots ,n_k},y)=\sum _{l=1}^k{n}_l(n_l-1)\prod _{j=1,j\ne l}^k{n}_j^2{y}^k.$$

If $n_1=n_2=\cdots =n_k=2$, then the Hamming graph is famous as a k-dimensional hypercube graph, and it is symbolized by $Q_k$ (see Figure 2). Then, ${\xi }^{ds}(Q_k,y)=2^{(2k-1)}k{y}^k$.

Rook’s graph is defined as the Cartesian product of $K_n$ and $K_m$ while an s-prism graph is defined as a Cartesian product $K_2\otimes C_s$. Rook’s graph $(K_3\otimes K_2)$ and the 6-prism graph are shown in Figure 3.

Example 6.

For an s-prism graph $K_2\otimes C_s$, the eccentric-distance sum polynomial is given by

$${\xi }^{ds}(K_2\otimes C_s,y)=\left\{\begin{array}{cc}2s^2\left(1+{\displaystyle \frac{s}{2}}\right)y^{(s+2)/2},\hfill & for s is even,\hfill \\ \\ s(s^2+2s-1)y^{(s+1)/2},\hfill & for s is odd.\hfill \end{array}\right.$$

Example 7.

The Cartesian product of $K_n$ and $K_m$ yields Rook’s graph. Then, the eccentric-distance sum polynomial is ${\xi }^{ds}(K_n\otimes K_m,y)=nm(2mn-m-n)y^2$.

4. Lexicographic Product

The lexicographic product of ${\mathcal{H}}_p$ and ${\mathcal{H}}_q$ graphs is represented by ${\mathcal{H}}_p\left[{\mathcal{H}}_q\right].$ It has $\mathcal{V}\left({\mathcal{H}}_p\right)\times \mathcal{V}\left({\mathcal{H}}_q\right)$ vertex set, and $(h_p,g_p)(h_q,g_q)\in \mathcal{E}\left({\mathcal{H}}_p\left[{\mathcal{H}}_q\right]\right)$ if $h_p{h}_q\in \mathcal{E}\left({\mathcal{H}}_p\right)$ or $h_p=h_q$ and $g_p{g}_q\in \mathcal{E}\left({\mathcal{H}}_q\right)$. The following lemma is very valuable in finding the eccentric-distance sum polynomial of ${\mathcal{H}}_p\left[{\mathcal{H}}_q\right]$. The closed fence graph $C_8\left[P_2\right]$ and Catlin graph $C_5\left[K_3\right]$ are depicted in Figure 4.

Lemma 3.

For graphs ${\mathcal{H}}_p$ and ${\mathcal{H}}_q$, with $|\mathcal{V}\left({\mathcal{H}}_l\right)|=n_l$ and $|\mathcal{E}\left({\mathcal{H}}_l\right)|=e_l$, where $l=p$ or q, we have

(a) 

$d_{{\mathcal{H}}_p\left[{\mathcal{H}}_q\right]}((h_p,g_p),(h_q,g_q))=\left\{\begin{array}{cc}{d}_{{\mathcal{H}}_p}(h_p,h_q),\hfill & if h_p\ne h_q,\hfill \\ 0,\hfill & if h_p=h_q,g_p=g_q,\hfill \\ 1,\hfill & if h_p=h_q,g_p{g}_q\in \mathcal{E}\left({\mathcal{H}}_q\right),\hfill \\ 2,\hfill & if h_p=h_q,g_p{g}_q\notin \mathcal{E}\left({\mathcal{H}}_q\right).\hfill \end{array}\right.$

(b) 

$e{c}_{{\mathcal{H}}_p\left[{\mathcal{H}}_q\right]}\left((h_p,g_p)\right)=\left\{\begin{array}{cc}e{c}_{{\mathcal{H}}_p}\left(h_p\right),\hfill & if h_p\notin \mathcal{N}\left({\mathcal{H}}_p\right),\hfill \\ 1,\hfill & if h_p\in \mathcal{N}\left({\mathcal{H}}_p\right),g_q\in \mathcal{N}\left({\mathcal{H}}_q\right),\hfill \\ 2,\hfill & if h_p\in \mathcal{N}\left({\mathcal{H}}_p\right),g_q\notin \mathcal{N}\left({\mathcal{H}}_q\right).\hfill \end{array}\right.$

Next, we find the eccentric-distance sum polynomial of a lexicographic product of two graphs with the help of Lemma 3.

Theorem 3.

Let ${\mathcal{H}}_p$ and ${\mathcal{H}}_q$ be graphs with $|\mathcal{V}\left({\mathcal{H}}_l\right)|=n_l$ and $|\mathcal{E}\left({\mathcal{H}}_l\right)|=e_l$, where $l=p$ or q. Then,

$$\begin{array}{ccc}\hfill {\xi }^{ds}({\mathcal{H}}_p\left[{\mathcal{H}}_q\right],y)& =& (n_1{n}_2-1)|\mathcal{N}\left({\mathcal{H}}_p\right)\left|\right|\mathcal{N}\left({\mathcal{H}}_q\right)|y+|\mathcal{N}\left({\mathcal{H}}_p\right)\left|\right(n_2(n_1+n_2-2)-2e_2\hfill \\ & -& |\mathcal{N}\left({\mathcal{H}}_q\right)|(n_1{n}_2-1))y^2+\left|\mathcal{N}\left({\mathcal{H}}_p\right)\right|(2e_2-n_2(n_1{n}_2+n_2+2))\hfill \\ & +& 2(n_2(n_2-1)-e_2)\tau ({\mathcal{H}}_p,y)+n_2^2{\xi }^{ds}({\mathcal{H}}_p,y).\hfill \end{array}$$

Proof. 

By Equation (1) and Lemma 3, we have:

$$\begin{array}{ccc}\hfill {\xi }^{ds}({\mathcal{H}}_p\left[{\mathcal{H}}_q\right],y)& =& \sum _{(h_p,g_p)\in \mathcal{V}\left({\mathcal{H}}_p\left[{\mathcal{H}}_q\right]\right)}\sum _{(h_q,g_q)\in \mathcal{V}\left({\mathcal{H}}_p\left[{\mathcal{H}}_q\right]\right)}{d}_{{\mathcal{H}}_p\left[{\mathcal{H}}_q\right]}((h_p,g_p),(h_q,g_q))y^{e{c}_{{\mathcal{H}}_p\left[{\mathcal{H}}_q\right]}\left((h_p,g_p)\right)}\hfill \\ & =& \sum _{h_p\in \mathcal{N}\left({\mathcal{H}}_p\right)}\sum _{g_p\in \mathcal{N}\left({\mathcal{H}}_q\right)}\sum _{g_q\in \mathcal{V}\left({\mathcal{H}}_q\right)\backslash \left\{g_p\right\}}\left(1\right)y^{\left(1\right)}\hfill \\ & +& \sum _{h_p\in \mathcal{N}\left({\mathcal{H}}_p\right)}\sum _{g_p\in \mathcal{N}\left({\mathcal{H}}_q\right)}\sum _{h_q\in \mathcal{V}\left({\mathcal{H}}_p\right)\backslash \left\{h_p\right\}}\sum _{g_q\in \mathcal{V}\left({\mathcal{H}}_q\right)}\left(1\right)y^{\left(1\right)}\hfill \\ & +& \sum _{h_p\in \mathcal{N}\left({\mathcal{H}}_p\right)}\sum _{g_p\in \mathcal{V}\left({\mathcal{H}}_q\right)\backslash \mathcal{N}\left({\mathcal{H}}_q\right)}\sum _{g_q\in {\gamma }_{{\mathcal{H}}_q}\left(g_p\right)}\left(1\right)y^{\left(2\right)}\hfill \\ & +& \sum _{h_p\in \mathcal{N}\left({\mathcal{H}}_p\right)}\sum _{g_p\in \mathcal{V}\left({\mathcal{H}}_q\right)\backslash \mathcal{N}\left({\mathcal{H}}_q\right)}\sum _{g_q\in \mathcal{V}\left({\mathcal{H}}_q\right)\backslash ({\gamma }_{{\mathcal{H}}_q}\left(g_p\right)\cup \left\{g_p\right\})}\left(2\right)y^{\left(2\right)}\hfill \\ & +& \sum _{h_p\in \mathcal{N}\left({\mathcal{H}}_p\right)}\sum _{g_p\in \mathcal{V}\left({\mathcal{H}}_q\right)\backslash \mathcal{N}\left({\mathcal{H}}_q\right)}\sum _{h_q\in \mathcal{V}\left({\mathcal{H}}_p\right)\backslash \left\{h_p\right\}}\sum _{g_q\in \mathcal{V}\left({\mathcal{H}}_q\right)}\left(1\right)y^{\left(2\right)}\hfill \\ & +& \sum _{h_p\in \mathcal{V}\left({\mathcal{H}}_p\right)\backslash \mathcal{N}\left({\mathcal{H}}_p\right)}\sum _{g_p\in \mathcal{V}\left({\mathcal{H}}_q\right)}\sum _{g_q\in {\gamma }_{{\mathcal{H}}_q}\left(g_p\right)}\left(1\right)y^{e{c}_{{\mathcal{H}}_p}\left(h_p\right)}\hfill \\ & +& \sum _{h_p\in \mathcal{V}\left({\mathcal{H}}_p\right)\backslash \mathcal{N}\left({\mathcal{H}}_p\right)}\sum _{g_p\in \mathcal{V}\left({\mathcal{H}}_q\right)}\sum _{g_q\in \mathcal{V}\left({\mathcal{H}}_q\right)\backslash ({\gamma }_{{\mathcal{H}}_q}\left(g_p\right)\cup \left\{g_p\right\})}2y^{e{c}_{{\mathcal{H}}_p}\left(h_p\right)}\hfill \\ & +& \sum _{h_p\in \mathcal{V}\left({\mathcal{H}}_p\right)\backslash \mathcal{N}\left({\mathcal{H}}_p\right)}\sum _{g_p\in \mathcal{V}\left({\mathcal{H}}_q\right)}\sum _{h_q\in \mathcal{V}\left({\mathcal{H}}_p\right)\backslash ({\gamma }_{{\mathcal{H}}_p}\left(h_p\right)\cup \left\{h_p\right\})}\hfill \\ & & \sum _{g_q\in \mathcal{V}\left({\mathcal{H}}_q\right)}{d}_{{\mathcal{H}}_p}(h_p,h_q)y^{e{c}_{{\mathcal{H}}_p}\left(h_p\right)}\hfill \\ & =& |\mathcal{N}\left({\mathcal{H}}_p\right)\left|\right|\mathcal{N}\left({\mathcal{H}}_q\right)|(n_2-1)y+|\mathcal{N}\left({\mathcal{H}}_p\right)\left|\right|\mathcal{N}\left({\mathcal{H}}_q\right)|n_2(n_1-1)y+\left|\mathcal{N}\left({\mathcal{H}}_p\right)\right|\hfill \\ & & [2e_2-|\mathcal{N}\left({\mathcal{H}}_q\right)|(n_2-1)]y^2+2|\mathcal{N}\left({\mathcal{H}}_p\right)\left|\right[(n_2-|\mathcal{N}\left({\mathcal{H}}_q\right)\left|\right)(n_2-1)-(2e_2\hfill \\ & -& |\mathcal{N}\left({\mathcal{H}}_q\right)|(n_2-1)\left)\right]y^2+|\mathcal{N}\left({\mathcal{H}}_p\right)|n_2(n_1-1)(n_2-|\mathcal{N}\left({\mathcal{H}}_q\right)\left|\right)y^2+2e_2\hfill \\ & & (\tau ({\mathcal{H}}_p,y)-|\mathcal{N}\left({\mathcal{H}}_p\right)\left|\right)+2[n_2(n_2-1)-2e_2](\tau ({\mathcal{H}}_p,y)-|\mathcal{N}\left({\mathcal{H}}_p\right)\left|\right)\hfill \\ & +& n_2^2[{\xi }^{ds}({\mathcal{H}}_p,y)-|\mathcal{N}\left({\mathcal{H}}_p\right)\left|(n_1-1)\right]\hfill \\ & =& (n_1{n}_2-1)|\mathcal{N}\left({\mathcal{H}}_p\right)\left|\right|\mathcal{N}\left({\mathcal{H}}_q\right)|y+|\mathcal{N}\left({\mathcal{H}}_p\right)\left|\right(n_2(n_1+n_2-2)-2e_2\hfill \\ & -& |\mathcal{N}\left({\mathcal{H}}_q\right)|(n_1{n}_2-1))y^2+\left|\mathcal{N}\left({\mathcal{H}}_p\right)\right|(2e_2-n_2(n_1{n}_2+n_2+2))\hfill \\ & +& 2(n_2(n_2-1)-e_2)\tau ({\mathcal{H}}_p,y)+n_2^2{\xi }^{ds}({\mathcal{H}}_p,y).\hfill \end{array}$$

After some simplifications, we get the required result. □

Example 8.

The closed fence graph is the lexicographic product of $C_l$ and $P_2$. Then, from Theorem 3 and Proposition 1, we have:

$${\xi }^{ds}(C_l\left[P_2\right],y)=\left\{\begin{array}{cc}6l{y}^{l/2},\hfill & for l is even,\hfill \\ 6l{y}^{(l-1)/2},\hfill & for l is odd.\hfill \end{array}\right.$$

Example 9.

For the Catlin graph $\left(C_5\left[K_3\right]\right)$, the eccentric-distance sum polynomial is $210y^2$.

5. Corona Product

The corona ${\mathcal{H}}_p\circ {\mathcal{H}}_q$ of graphs ${\mathcal{H}}_p$ and ${\mathcal{H}}_q$ is constructed by drawing a copy of ${\mathcal{H}}_p$ and $|\mathcal{V}\left({\mathcal{H}}_p\right)|=n_p$ copies of ${\mathcal{H}}_q$ and linking the k-th vertex of ${\mathcal{H}}_p$ to every vertex of k-th copy of ${\mathcal{H}}_q$, where $1\le k\le n_p$. The following lemma gives some information that will be helpful in our main result.

Lemma 4.

Let ${\mathcal{H}}_p$ and ${\mathcal{H}}_q$ be graphs with $|\mathcal{V}\left({\mathcal{H}}_l\right)|=n_l$ and $|\mathcal{E}\left({\mathcal{H}}_l\right)|=e_l$, where $l=p,q$. Then, we have the following:

(a) 

$d_{{\mathcal{H}}_p\circ {\mathcal{H}}_q}(h_p,g_p)=\left\{\begin{array}{cc}{d}_{{\mathcal{H}}_p}(h_p,g_p),\hfill & if h_p,g_p\in \mathcal{V}\left({\mathcal{H}}_p\right),\hfill \\ d_{{\mathcal{H}}_p}(h_p,w_m)+1,\hfill & if h_p\in \mathcal{V}\left({\mathcal{H}}_p\right)and{g}_p\in \mathcal{V}\left({\mathcal{H}}_{2,m}\right),1\le m\le n,\hfill \\ 0,\hfill & if h_p=g_p\in \mathcal{V}\left({\mathcal{H}}_{2,m}\right),1\le m\le n,\hfill \\ 1,\hfill & if h_p{g}_p\in \mathcal{E}\left({\mathcal{H}}_{2,m}\right),1\le m\le n,\hfill \\ 2,\hfill & if h_p,g_p\in \mathcal{V}\left({\mathcal{H}}_{2,m}\right),h_p{g}_p\notin \mathcal{E}\left({\mathcal{H}}_{2,m}\right),\hfill \\ & h_p\ne g_p,1\le m\le n,\hfill \\ d_{{\mathcal{H}}_p}(w_m,w_j)+2,\hfill & if h_p\in \mathcal{V}\left({\mathcal{H}}_p\right),g_p\in \mathcal{V}\left({\mathcal{H}}_{2,m}\right),1\le m\le n,\hfill \end{array}\right.$

(b) 

$e{c}_{{\mathcal{H}}_p\circ {\mathcal{H}}_q}\left(h_p\right)=\left\{\begin{array}{cc}e{c}_{{\mathcal{H}}_p}\left(h_p\right)+1,\hfill & if h_p\in \mathcal{V}\left({\mathcal{H}}_p\right),\hfill \\ e{c}_{{\mathcal{H}}_p}\left(w_m\right)+2,\hfill & if h_p\in \mathcal{V}\left({\mathcal{H}}_{2,m}\right),1\le m\le n,\hfill \end{array}\right.$ where ${\mathcal{H}}_{2,m}$ is the m-th copy of ${\mathcal{H}}_q$.

In Theorem 4, we find out the eccentric-distance sum polynomial of ${\mathcal{H}}_p\circ {\mathcal{H}}_q$.

Theorem 4.

Let ${\mathcal{H}}_p$ and ${\mathcal{H}}_q$ be graphs with $|\mathcal{V}\left({\mathcal{H}}_l\right)|=n_l$ and $|\mathcal{E}\left({\mathcal{H}}_l\right)|=e_l$, where $l=p,q$. Then

$$\begin{array}{ccc}\hfill {\xi }^{ds}({\mathcal{H}}_p\circ {\mathcal{H}}_q,y)& =& y(1+n_2)(1+n_{2}y){\xi }^{ds}({\mathcal{H}}_p,y)+y(n_1{n}_2(1+y)+2n_{2}y(n_1{n}_2-1)\hfill \\ & -& 2ey)\tau ({\mathcal{H}}_p,y).\hfill \end{array}$$

Proof. 

Using Equation (1) and Lemma 4, we have:

$$\begin{array}{ccc}\hfill {\xi }^{ds}({\mathcal{H}}_p\circ {\mathcal{H}}_q,y)& =& \sum _{h\in \mathcal{V}({\mathcal{H}}_p\circ {\mathcal{H}}_q)}\sum _{g\in \mathcal{V}({\mathcal{H}}_p\circ {\mathcal{H}}_q)}{d}_{{\mathcal{H}}_p\circ {\mathcal{H}}_q}(h,g)y^{e{c}_{{\mathcal{H}}_p\circ {\mathcal{H}}_q}\left(h\right)}\hfill \\ & =& \sum _{h\in \mathcal{V}\left({\mathcal{H}}_p\right)}\sum _{g\in \mathcal{V}\left({\mathcal{H}}_p\right)}{d}_{{\mathcal{H}}_p}(h,g)y^{e{c}_{{\mathcal{H}}_p}\left(h\right)+1}+\sum _{l=1}^n\sum _{h\in \mathcal{V}\left({\mathcal{H}}_p\right)}\sum _{g\in \mathcal{V}\left({\mathcal{H}}_{2,i}\right)}(d_{{\mathcal{H}}_p}(h,w_l)\hfill \\ & +& 1)y^{e{c}_{{\mathcal{H}}_p}\left(h\right)+1}+\sum _{l=1}^n\sum _{h\in \mathcal{V}\left({\mathcal{H}}_{2,l}\right)}\sum _{g\in \mathcal{V}\left({\mathcal{H}}_p\right)}(d_{{\mathcal{H}}_p}(g,w_l)+1)y^{e{c}_{{\mathcal{H}}_p}\left(w_l\right)+2}\hfill \\ & +& \sum _{l=1}^n\sum _{j=1,j\ne l}^n\sum _{h\in \mathcal{V}\left({\mathcal{H}}_{2,l}\right)}\sum _{g\in \mathcal{V}\left({\mathcal{H}}_{2,l}\right)}(d_{{\mathcal{H}}_p}(w_l,w_j)+2)y^{e{c}_{{\mathcal{H}}_p}\left(w_l\right)+2}\hfill \\ & +& \sum _{l=1}^n\sum _{h\in \mathcal{V}\left({\mathcal{H}}_{2,i}\right)}\sum _{g\in {\gamma }_{{\mathcal{H}}_{2,l}}\left(h\right)}\left(1\right)y^{e{c}_{{\mathcal{H}}_p}\left(w_l\right)+2}\hfill \\ & +& \sum _{l=1}^n\sum _{h\in \mathcal{V}\left({\mathcal{H}}_{2,l}\right)}\sum _{g\in \mathcal{V}\left({\mathcal{H}}_{2,l}\right)\backslash ({\gamma }_{{\mathcal{H}}_{2,l}}\left(h\right)\cup \left\{h\right\})}\left(2\right)y^{e{c}_{{\mathcal{H}}_p}\left(w_l\right)+2}\hfill \\ & =& {\xi }^{ds}({\mathcal{H}}_p,y)y+n_2{\xi }^{ds}({\mathcal{H}}_p,y)y+n_1{n}_2\tau ({\mathcal{H}}_p,y)y+n_2{\xi }^{ds}({\mathcal{H}}_p,y)y^2\hfill \\ & +& n_1{n}_2\tau ({\mathcal{H}}_p,y)y^2+n_2^2{\xi }^{ds}({\mathcal{H}}_p,y)y^2+2n_2^2(n_1-1)\tau ({\mathcal{H}}_p,y)y^2\hfill \\ & +& 2e_2\tau ({\mathcal{H}}_p,y)y^2+2(n_2(n_2-1)-2e_2)\tau ({\mathcal{H}}_p,y)y^2.\hfill \end{array}$$

After some simple computation steps, we can get the desired expression. This finishes the proof. □

For a graph $\mathcal{H}$, the t-thorny graph is gained by linking t-number of pendant vertices to each vertex of $\mathcal{H}$, and it is represented by ${\mathcal{H}}^t$. The t-thorny graph of $\mathcal{H}$ is specified as $\mathcal{H}\circ {\overline{K}}_t$ (see Figure 5).

Thus, from Theorem 4, the next corollary follows.

Corollary 3.

For a graph $\mathcal{H}$ with $\left|\mathcal{E}\right(\mathcal{H}\left)\right|=e$ and $\left|\mathcal{V}\right(\mathcal{H}\left)\right|=n$, we have:

$${\xi }^{ds}(\mathcal{H}\circ \overline{K_t},y)=y(1+t)(1+ty){\xi }^{ds}(\mathcal{H},y)+y(tn(1+y)+2ty(nt-1))\tau (\mathcal{H},y).$$

Example 10.

The bottleneck graph $\mathcal{B}$ (see Figure 6) of $\mathcal{H}$ is obtained by taking the corona product of $K_2$ with the given n vertex graph $\mathcal{H}$. Then ${\xi }^{ds}(\mathcal{B},y)=2y^2(1+y(n+2)+y(3n^2-e))$, where $\left|\mathcal{E}\right(\mathcal{H}\left)\right|=e$.

6. Generalized Hierarchical Product

Barrire et al. [21] brought in a generalized hierarchical product that is a continuation of the (standard) hierarchical and Cartesian product [22]. Let ${\mathcal{H}}_p$ and ${\mathcal{H}}_q$ be graphs with $\mathcal{A}\subseteq \mathcal{V}\left({\mathcal{H}}_p\right)$, where $\mathcal{A}\ne \varnothing$. Then, the generalized hierarchical product ${\mathcal{H}}_p\left(\mathcal{A}\right)\sqcap {\mathcal{H}}_q$ is a graph with vertex set $\mathcal{V}\left({\mathcal{H}}_p\right)\times \mathcal{V}\left({\mathcal{H}}_q\right)$ and $(h_p,g_p)(h_q,g_a)\in \mathcal{E}({\mathcal{H}}_p\left(\mathcal{A}\right)\sqcap {\mathcal{H}}_q)$ if $h_p=h_q\in \mathcal{A}$ and $g_p{g}_q\in \mathcal{E}\left({\mathcal{H}}_q\right)$ or $h_p{h}_q\in \mathcal{E}\left({\mathcal{H}}_p\right)$ and $g_p=g_q$. For $\mathcal{A}\subseteq \mathcal{V}\left({\mathcal{H}}_p\right)$, where $\mathcal{A}\ne \varnothing$, a path among vertices $h,g\in \mathcal{V}\left({\mathcal{H}}_p\right)$ through $\mathcal{A}$ is an $hg$-path in ${\mathcal{H}}_p$ consisting of some vertex $w\in \mathcal{A}$ ($w=h$ or $w=g$). The distance among h and g through $\mathcal{A}$, symbolized by $d_{{\mathcal{H}}_p\left(\mathcal{A}\right)}(h,g)$, is the length of a minimum path among h and g through $\mathcal{A}$. It is easy to see that if $h\in \mathcal{A}$ and $g\in \mathcal{A}$, then $d_{{\mathcal{H}}_p\left(\mathcal{A}\right)}(h,g)=d_{{\mathcal{H}}_p}(h,g)$. For more details, see [23]. The upcoming important lemma gives some properties of ${\mathcal{H}}_p\left(\mathcal{A}\right)\sqcap {\mathcal{H}}_q$.

Lemma 5.

Let ${\mathcal{H}}_p$ and ${\mathcal{H}}_q$ be graphs with $|\mathcal{V}\left({\mathcal{H}}_l\right)|=n_l$ and $|\mathcal{E}\left({\mathcal{H}}_l\right)|=e_l$, where $l=p$ or q. Then, we have the following:

(a) 

$d_{{\mathcal{H}}_p\left(\mathcal{A}\right)\sqcap {\mathcal{H}}_q}((h_p,g_p),(h_q,g_q))=\left\{\begin{array}{cc}{d}_{{\mathcal{H}}_p}(h_p,h_q)+d_{{\mathcal{H}}_q}(g_p,g_q),\hfill & if g_p\ne g_q,\hfill \\ d_{{\mathcal{H}}_p}(h_p,h_q)\hfill & if g_p=g_q.\hfill \end{array}\right.$

(b) 

$e{c}_{{\mathcal{H}}_p\left(\mathcal{A}\right)\sqcap {\mathcal{H}}_q}(h_p,g_q)=e{c}_{{\mathcal{H}}_p\left(\mathcal{A}\right)}\left(h_p\right)+e{c}_{{\mathcal{H}}_q}\left(g_q\right)$.

In the next theorem, we work out the eccentric-distance sum polynomial of ${\mathcal{H}}_p\left(U\right)\sqcap {\mathcal{H}}_q$ in the form of eccentric-distance sum and total eccentricity polynomials of ${\mathcal{H}}_l$, where $l=p,q$.

Theorem 5.

Let ${\mathcal{H}}_p$ and ${\mathcal{H}}_q$ be graphs with $|\mathcal{V}\left({\mathcal{H}}_l\right)|=n_l$, $|\mathcal{E}\left({\mathcal{H}}_l\right)|=e_l$, $l=p$ or q, and $\mathcal{A}\subseteq \mathcal{V}\left({\mathcal{H}}_p\right)$. Then, we obtain

$${\xi }^{ds}({\mathcal{H}}_p\left(\mathcal{A}\right)\sqcap {\mathcal{H}}_q,y)=n_2{\xi }^{ds}({\mathcal{H}}_p\left(\mathcal{A}\right),y)\tau ({\mathcal{H}}_q,y)+n_1\tau ({\mathcal{H}}_p\left(\mathcal{A}\right),y){\xi }^{ds}({\mathcal{H}}_q,y).$$

Proof. 

Using Equation (1) and Lemma 5, we get:

$$\begin{array}{ccc}\hfill {\xi }^{ds}({\mathcal{H}}_p\left(\mathcal{A}\right)\sqcap {\mathcal{H}}_q,y)& =& \sum _{(h_p,g_p)\in \mathcal{V}({\mathcal{H}}_p\left(\mathcal{A}\right)\sqcap {\mathcal{H}}_q)}\sum _{(h_q,g_q)\in \mathcal{V}({\mathcal{H}}_p\left(\mathcal{A}\right)\sqcap {\mathcal{H}}_q)}{d}_{{\mathcal{H}}_p\left(\mathcal{A}\right)\sqcap {\mathcal{H}}_q}((h_p,g_p),(h_q,g_q))\hfill \\ & & y^{e{c}_{{\mathcal{H}}_p\left(\mathcal{A}\right)\sqcap {\mathcal{H}}_q}\left((h_p,g_p)\right)}\hfill \\ & =& \sum _{h_p,h_q\in \mathcal{V}\left({\mathcal{H}}_p\right)}\sum _{g_p,g_q\in \mathcal{V}\left({\mathcal{H}}_q\right),g_q\ne g_p}(d_{{\mathcal{H}}_p\left(\mathcal{A}\right)}(h_p,h_q)+d_{{\mathcal{H}}_q}(g_p,g_q))\hfill \\ & & y^{(e{c}_{{\mathcal{H}}_p\left(\mathcal{A}\right)}\left(h_p\right)+e{c}_{{\mathcal{H}}_q}\left(g_p\right))}+\sum _{h_p,h_q\in \mathcal{V}\left({\mathcal{H}}_p\right)}\sum _{g_p,g_q\in \mathcal{V}\left({\mathcal{H}}_q\right),g_q=g_p}{d}_{{\mathcal{H}}_p}(h_p,h_q)\hfill \\ & & y^{(e{c}_{{\mathcal{H}}_p\left(\mathcal{A}\right)}\left(h_p\right)+e{c}_{{\mathcal{H}}_q}\left(g_p\right))}\hfill \\ & =& \sum _{h_p\in \mathcal{V}\left({\mathcal{H}}_p\right)}\sum _{h_q\in \mathcal{V}\left({\mathcal{H}}_p\right)}{d}_{{\mathcal{H}}_p\left(\mathcal{A}\right)}(h_p,h_q)y^{e{c}_{{\mathcal{H}}_p\left(\mathcal{A}\right)}\left(h_p\right)}\sum _{g_q\in \mathcal{V}\left({\mathcal{H}}_q\right),g_q\ne g_p}\hfill \\ & & \sum _{g_p\in \mathcal{V}\left({\mathcal{H}}_q\right)}{y}^{e{c}_{{\mathcal{H}}_q}\left(g_p\right)}+\sum _{h_q\in \mathcal{V}\left({\mathcal{H}}_p\right)}\sum _{h_p\in \mathcal{V}\left({\mathcal{H}}_p\right)}{y}^{e{c}_{{\mathcal{H}}_p\left(\mathcal{A}\right)}\left(h_p\right)}\sum _{g_p\in \mathcal{V}\left({\mathcal{H}}_q\right)}\hfill \\ & & \sum _{g_q\in \mathcal{V}\left({\mathcal{H}}_q\right),g_q\ne g_p}{d}_{{\mathcal{H}}_q}(g_p,g_q)y^{e{c}_{{\mathcal{H}}_q}\left(g_p\right)}+\sum _{h_p\in \mathcal{V}\left({\mathcal{H}}_p\right)}\sum _{h_q\in \mathcal{V}\left({\mathcal{H}}_p\right)}{d}_{{\mathcal{H}}_p}(h_p,h_q)y^{e{c}_{{\mathcal{H}}_p\left(\mathcal{A}\right)}\left(h_p\right)}\hfill \\ & & \sum _{g_p\in \mathcal{V}\left({\mathcal{H}}_q\right)}\sum _{g_q\in \mathcal{V}\left({\mathcal{H}}_q\right),g_q=g_p}{y}^{e{c}_{{\mathcal{H}}_q}\left(g_p\right)}\hfill \\ & =& n_2{\xi }^{ds}({\mathcal{H}}_p\left(\mathcal{A}\right),y)\tau ({\mathcal{H}}_q,y)+n_1\tau ({\mathcal{H}}_p\left(\mathcal{A}\right),y){\xi }^{ds}({\mathcal{H}}_q,y).\hfill \end{array}$$

This finishes the proof. □

7. Conclusions

The numerical description of molecular graphs by using graph products and graph invariants is a very important graph theory implementation. This graph theory implementation may be in the form of atomic, spectra, polynomials or molecular topological invariants. However, the characterization of graphs by polynomials is a new line of research in modern graph theory. In this paper, we derived the expressions for eccentric-distance sum polynomial of some graph products such as join, Cartesian, lexicographic, corona and generalized hierarchical products. Moreover, we applied our outcomes to calculate this polynomial for some significant families of molecular graphs in the form of the above graph products. Some topological indices and polynomials are still unknown for different graph operations, derived graphs, graph structures, etc. Topological indices and polynomials for more graph operations and structures are worth studying in the future.