Computing Degree-Based Topological Indices of Jahangir Graph

1. Introduction

The study of topological indices, based on distance in a graph, was effectively employed in 1947, in chemistry by Weiner [1]. He introduced a distance-based topological index called the Wiener index to correlate properties of alkenes and the structures of their molecular graphs.

Topological indices play a vital role in computational and theoretical aspects of chemistry in predicting material properties [2, 3, 4, 5, 6, 7, 8]. Several algebraic polynomials have useful applications in chemistry [9, 10]. A graph \(G\) is an ordered pair \((V,E)\), where \(V\) is the set of vertices and \(E\) is the set of edges. A path from a vertex \(v\) to a vertex \(w\) is a sequence of vertices and edges that starts from \(v\) and stops at \(w\). The number of edges in a path is called the length of that path.

A graph is said to be connected if there is a path between any two of its vertices. The distance \(d(u,v)\) between two vertices \(u\), \(v\) of a connected graph \(G\) is the length of a shortest path between them. Graph theory is contributing a lion’s share in many areas such as chemistry, physics, pharmacy, as well as in industry [11]. We will start with some preliminary facts.

The first and second multiplicative Zagreb indices [12] are defined as \begin{equation} MZ_{1}(G)=\prod\limits_{u\in V(G)}(d_{u})^{2}, \end{equation} \begin{equation} MZ_{2}(G)=\prod\limits_{uv\in E(G)}d_{u}. d_{u}, \end{equation} and the Narumi-Kataymana index [13] is defined as \begin{equation} NK(G)=\prod\limits_{u\in V(G)}d_{u}, \end{equation} Like the Wiener index, these types of indices are the focus of considerable research in computational chemistry [14, 15, 16, 17]. For example, in 2011, Gutman [14] characterized the multiplicative Zagreb indices for trees and determined the unique trees that obtained maximum and minimum values for \(M_{1}(G)\) and \(M_{2}(G)\). Wang et al. [17] define the following index for k-trees, \begin{equation} W^{s}_{1}(G)=\prod\limits_{u\in V(G)}(d_{u})^{s}. \end{equation} Notice that \(s=1,2\) is the Narumi-Katayama and Zagreb index, respectively. Based on the successful consideration of multiplicative Zagreb indices, Eliasi et al. [18] continued to define a new multiplicative version of the first Zagreb index as \begin{equation} MZ^{\ast}_{1}(G)=\prod\limits_{uv\in E(G)}(d_{u}+d_{u}), \end{equation} Furthering the concept of indexing with the edge set, the first author introduced the first and second hyper-Zagreb indices of a graph [19]. They are defined as \begin{equation} HII_{1}(G)=\prod\limits_{uv\in E(G)}(d_{u}+d_{u})^{2}, \end{equation} \begin{equation} HII_{2}(G)=\prod\limits_{uv\in E(G)}(d_{u}.d_{u})^{2}, \end{equation} In [20] Kulli et al. defined the first and second generalized Zagreb indices \begin{equation} MZ^{a}_{1}(G)=\prod\limits_{uv\in E(G)}(d_{u}+d_{u})^{a}, \end{equation} \begin{equation} MZ^{a}_{2}(G)=\prod\limits_{uv\in E(G)}(d_{u}.d_{u})^{a}, \end{equation} Multiplicative sum connectivity and multiplicative product connectivity indices [21] are define as: \begin{equation} SCII(G)=\prod\limits_{uv\in E(G)}\frac{1}{(d_{u}+d_{u})}, \end{equation} \begin{equation} PCII(G)(G)=\prod\limits_{uv\in E(G)}\frac{1}{(d_{u}.d_{u})}, \end{equation} Multiplicative atomic bond connectivity index and multiplicative Geometric arithmetic index are defined as \begin{equation} ABCII(G)=\prod\limits_{uv\in E(G)}\sqrt{\frac{d_{u}+d_{u}-2}{d_{u}.d_{u}}}, \end{equation} \begin{equation} GAII(G)=\prod\limits_{uv\in E(G)}\frac{2\sqrt{d_{u}.d_{u}}}{d_{u}+d_{u}}, \end{equation} \begin{equation} GA^{a}II(G)=\prod\limits_{uv\in E(G)}\left(\frac{2\sqrt{d_{u}.d_{u}}}{d_{u}+d_{u}}\right)^{a} \end{equation} In this paper we compute multiplicative indices of Jahangir graphs. The Jahangir graph \(J_{m,n}\) is a graph on \(nm + 1\) vertices and \(m(n + 1)\) edges for all \(n\geq 2\) and \(m\geq 3\). \(J_{m,n}\) consists of a cycle \(C_{mn}\) with one additional vertex which is adjacent to \(m\) vertices of \(C_{nm}\) at distance to each other. Figure 1 shows some particular cases of \(J_{m,n}\).

2. Computational Results

In this section, we present our computational results.

Theorem 2.1. Let \(J_{m,n}\) be the jahangir’s graph. Then

1. \(MZ^{a}_{1}(J_{m,n})=(4)^{am(n-2)}\times (5)^{2am}\times (3+m)^{am}\),
2. \(MZ^{a}_{2}(J_{m,n})= (4)^{am(n-2)}\times (6)^{2am}\times (3m)^{am}\),
3. \(G^{a}AII(J_{m,n})=\left(\frac{2\sqrt{6}}{5}\right)^{2am}\times \left(\frac{2\sqrt{3\times m}}{3+m}\right)^{am}\).

Proof. Let \(G\) be the graph of \(J_{m,n}\). It is clear that the total number of vertices in \(J_{m,n}\) are \(8n+2\) and total number of edges are \(10n+1\) The edge set of \(J_{m,n}\) has following three partitions, $$E_{1}=E_{2,2}=\{e=uv\in E(J_{m,n}): d_{u}=2, d_{v}=2\},$$ $$E_{1}=E_{2,3}=\{e=uv\in E(J_{m,n}): d_{u}=2, d_{v}=3\},$$ and $$E_{1}=E_{3,m}=\{e=uv\in E(J_{m,n}): d_{u}=3, d_{v}=m\}.$$ Now, $$\mid E_{1}(J_{m,n})\mid=m(n-2),$$ $$\mid E_{2}(J_{m,n})\mid=2m,$$ and $$\mid E_{1}(J_{m,n})\mid=m.$$

(1) \begin{eqnarray*} MZ^{a}_{1}(J_{m,n})&=& \prod\limits_{uv\in E(G)}(d_{u}+d_{v})^{a}\\ &=& \prod\limits_{uv\in E_{1}(J_{m,n})}(d_{u}+d_{v})^{a}+\prod\limits_{uv\in E_{2}(J_{m,n})}(d_{u}+d_{v})^{a}+\prod\limits_{uv\in E_{3}(J_{m,n})}(d_{u}+d_{v})^{a}\\ &=&(d_{u}+d_{v})^{a|E_{1}(J_{m,n})|}+(d_{u}+d_{v})^{a|E_{2}(J_{m,n})|}+(d_{u}+d_{v})^{a|E_{3}(J_{m,n})|}\\ &=& (2+2)^{am(n-2)}+(2+3)^{a(2m)}+(3+m)^{am}\\ &=&(4)^{am(n-2)}\times (5)^{2am}\times (3+m)^{am}. \end{eqnarray*} (2) \begin{eqnarray*} MZ^{a}_{2}(J_{m,n})&=& \prod\limits_{uv\in E(G)}(d_{u}.d_{v})^{a}\\ &=& \prod\limits_{uv\in E_{1}(J_{m,n})}(d_{u}.d_{v})^{a}+\prod\limits_{uv\in E_{2}(J_{m,n})}(d_{u}.d_{v})^{a}+\prod\limits_{uv\in E_{3}(J_{m,n})}(d_{u}.d_{v})^{a}\\ &=&(d_{u}.d_{v})^{a|E_{1}(J_{m,n})|}+(d_{u}.d_{v})^{a|E_{2}(J_{m,n})|}+(d_{u}.d_{v})^{a|E_{3}(J_{m,n})|}\\ &=& (2.2)^{am(n-2)}+(2.3)^{a(2m)}+(3.m)^{am}\\ &=&(4)^{am(n-2)}\times (6)^{2am}\times (3m)^{am}. \end{eqnarray*} (3) \begin{eqnarray*} G^{a}AII(J_{m,n})&=& \prod\limits_{uv\in E(G)}\left(\frac{2\sqrt{d_{u}d_{v}}}{d_{u}+d_{v}}\right)^{a}\\ &=& \prod\limits_{uv\in E(G)}\left(\frac{2\sqrt{d_{u}d_{v}}}{d_{u}+d_{v}}\right)^{a}+\prod\limits_{uv\in E(G)}\left(\frac{2\sqrt{d_{u}d_{v}}}{d_{u}+d_{v}}\right)^{a}\\ &&+\prod\limits_{uv\in E(G)}\left(\frac{2\sqrt{d_{u}d_{v}}}{d_{u}+d_{v}}\right)^{a}\\ &=&\left(\frac{2\sqrt{d_{u}d_{v}}}{d_{u}+d_{v}}\right)^{a|E_{1}(J_{m,n})|}\times \left(\frac{2\sqrt{d_{u}d_{v}}}{d_{u}+d_{v}}\right)^{a|E_{2}(J_{m,n})|}\\ &&\times \left(\frac{2\sqrt{d_{u}d_{v}}}{d_{u}+d_{v}}\right)^{a|E_{3}(J_{m,n})|}\\ &=&\left(\frac{2\sqrt{2.2}}{2+2}\right)^{am(n-2)}\times \left(\frac{2\sqrt{2.3}}{2+3}\right)^{a(2m)}\\ &&\times \left(\frac{2\sqrt{3.m}}{3+m}\right)^{am}\\ &=&\left(\frac{2\sqrt{6}}{5}\right)^{2am}\times \left(\frac{2\sqrt{3\times m}}{3+m}\right)^{am}. \end{eqnarray*}

Corollary 2.2. Let \(J_{m,n}\) be the Jahangir’s graph. Then

1. \(MZ_{1}(J_{m,n})=(4)^{m(n-2)}\times (5)^{2m}\times (3+m)^{m}\),
2. \(MZ_{2}(J_{m,n})= (4)^{m(n-2)}\times (6)^{2m}\times (3m)^{m}\),
3. \(GAII(J_{m,n})=\left(\frac{2\sqrt{6}}{5}\right)^{2m}\times \left(\frac{2\sqrt{3\times m}}{3+m}\right)^{m}\).

Proof. We get our result by putting \(\alpha=1\) in the Theorem 2.1.

Corollary 2.3. Let \(J_{m,n}\) be the Jahangir’s graph. Then

1. \(H II_{1}(J_{m,n})=(4)^{2m(n-2)}\times (5)^{4m}\times (3+m)^{2m}\),
2. \(H II_{2}(J_{m,n})= (4)^{2m(n-2)}\times (6)^{4m}\times (3m)^{am}\).

Proof. We get our desired results by putting \(\alpha=2\) in Theorem 2.1.

Corollary 2.4. Let \(J_{m,n}\) be the Jahangir’s graph. Then

1. \(X II(J_{m,n})=\left(\frac{1}{2}\right)^{m(n-2)}\times \left(\frac{1}{\sqrt{5}}\right)^{2m}\times \left(\frac{1}{\sqrt{m+1}}\right)^{mn}\),
2. \(\chi II(J_{m,n})=\left(\frac{1}{2}\right)^{m(n-2)}\times \left(\frac{1}{\sqrt{6}}\right)^{2m}\times \left(\frac{1}{\sqrt{m+1}}\right)^{mn}\).

Proof. We get our desired results by putting \(\alpha=\frac{-1}{2}\) in Theorem 2.1.

Theorem 2.5. Let \(J_{m,n}\) be the Jahangir’s graph. Then $$ABCII(J_{m,n})=\left(\frac{1}{\sqrt{2}}\right)^{mn}\times \left(\sqrt{\frac{m+1}{3m}}\right)^{mn}.$$

Proof. By using the edge partition of Jahangir’s graph given in Theorem 2.1. \begin{eqnarray*} ABCII(J_{m,n})&=&\prod\limits_{uv\in E(J_{m,n})}\sqrt{\frac{d_{u}+d_{u}-2}{d_{u}.d_{u}}}\\ &=&\prod\limits_{uv\in E_{1}(J_{m,n})}\sqrt{\frac{d_{u}+d_{u}-2}{d_{u}.d_{u}}}\times \prod\limits_{uv\in E_{2}(J_{m,n})}\sqrt{\frac{d_{u}+d_{u}-2}{d_{u}.d_{u}}}\\ &&\times \prod\limits_{uv\in E_{3}(J_{m,n})}\sqrt{\frac{d_{u}+d_{u}-2}{d_{u}.d_{u}}}\\ &=&\left(\sqrt{\frac{d_{u}+d_{u}-2}{d_{u}.d_{u}}}\right)^{|E_{1}(J_{m,n})|}\times \left(\sqrt{\frac{d_{u}+d_{u}-2}{d_{u}.d_{u}}}\right)^{|E_{2}(J_{m,n})|}\\ &&\times \left(\sqrt{\frac{d_{u}+d_{u}-2}{d_{u}.d_{u}}}\right)^{|E_{3}(J_{m,n})|}\\ &=&\left(\sqrt{\frac{d_{u}+d_{u}-2}{d_{u}.d_{u}}}\right)^{m(n-2)}\times \left(\sqrt{\frac{d_{u}+d_{u}-2}{d_{u}.d_{u}}}\right)^{2m}\\ &&\times \left(\sqrt{\frac{d_{u}+d_{u}-2}{d_{u}.d_{u}}}\right)^{m}\\ &=&\left(\sqrt{\frac{1}{2}}\right)^{m(n-2)}\times \left(\sqrt{\frac{1}{2}}\right)^{2m}\times \left(\sqrt{\frac{m+1}{3m}}\right)^{mn} \\ &=& \left(\frac{1}{\sqrt{2}}\right)^{mn}\times \left(\sqrt{\frac{m+1}{3m}}\right)^{mn} \end{eqnarray*}

Competing Interests

The authors declare that they have no competing interests.