Distancebased topological polynomials and indices of friendship graphs
 Wei Gao^{1},
 Mohammad Reza Farahani^{2},
 Muhammad Imran^{3, 4}Email author and
 M. R. Rajesh Kanna^{5}
Received: 26 April 2016
Accepted: 8 September 2016
Published: 15 September 2016
Abstract
Drugs and chemical compounds are often modeled as graphs in which the each vertex of the graph expresses an atom of molecule and covalent bounds between atoms are represented by the edges between their corresponding vertices. The topological indicators defined over this molecular graph have been shown to be strongly correlated to various chemical properties of the compounds. In this article, by means of graph structure analysis, we determine several distance based topological indices of friendship graph \( F_{3}^{(n)} \) which is widely appeared in various classes of new nanomaterials, drugs and chemical compounds.
Keywords
Mathematics Subject Classification
Background
Recent years have witnessed the rapid development in nanomaterials and drugs, which keeps in pace with the development of pharmacopedia. Because of the new issues raised by it, there is a need to test the physical, chemical and biological properties of these new chemical compounds, nanomaterials and drugs, which make the researchers’ workload increased much. Besides, to guarantee the effective results of the research, enough adequate equipment, reagents and human resources are needed to test the performance and the side effects of presented chemical compounds, nanomaterials and drugs. Nevertheless, funds in developing countries (like some countries in Southeast Asia, South America and Africa) are unable to afford the relevant equipment and reagents. Thanks to the contributions from the previous research which discovered that the chemical characteristics of chemical compounds, nanomaterials and drugs and their molecular structures are closely related. Simply speaking, it would benefit the medical and pharmaceutical scientists by providing supports to understand the properties of these chemical compounds, nanomaterials and drugs, if we learn their indicators based on the topological indices. This also helps to make up the experiments shortages. In this way, it can be predicted that the techniques on topological index computation will be welcomed by developing countries by providing the medical and biological information of new chemical compounds, nanomaterials and drugs without chemical experiment conditions.
In the graph computation model, the structure of chemical compounds, nanomaterials and drugs are described as a graph. Every atom are described by an individual vertex, and the chemical bond among them described by the edge. Let G be a graph which corresponds to a chemical structure with atom (vertex) set \( V(G) \) and chemical bond (edge) set \( E(G) \). The distance between vertices u and v, \( d_{G} (u,v) \) or \( d(u,v) \), in a graph is the number of edges in a shortest path connecting them and the diameter of a graph G, \( D(G) \) is the longest topological distance in G. The degree \( d_{v} (G) \) or d _{ v } of a vertex \( v \in V(G) \) is the number of vertices of G adjacent to v. A vertex \( v \in V(G) \) is said to be isolated, pendent, or fully connected if \( d_{v} = 0 \); \( d_{v} = 1 \), or \( d_{v} = n  1 \), respectively.
A topological index can be described as a realvalued map \( f:G \to R^{ + } \) which maps each chemical structure to certain real numbers. For decades, in order to test the features of chemical molecules, some indices like PI index, Wiener index, Harmonic index and Zagreb index are proposed. Meanwhile, some papers also devote to computing these topological indices of special molecular graph in chemical and pharmaceutical engineering.
In references (Polansky and Bonchev 1986; Sridhara et al. 2015; Gao et al. 2016a, b; Gao and Farahani 2016; Schultz 1989; Muller et al. 1990; Gutman and Polansky 1986; Trinajstic 1993; Klavžar and Gutman 1996; Gutman and Klavžar 1997; Hua 2009; Deng 2007; Chen et al. 2008; Zhou 2006), some properties and more historical details of the Wiener and Hyper Wiener indices and the Hosoya polynomial of molecular graphs are studded.
For more details on applications and mathematical properties of this topological based structure descriptor (the Wiener and HyperWiener indices and the Hosoya polynomial) see paper series (Wiener 1948; Randić 1993; Randić et al. 1994; Hosoya 1989; Polansky and Bonchev 1986; Sridhara et al. 2015; Gao et al. 2016a, b; Gao and Farahani 2016).
The friendship graph is the graph obtained by taking n copies of the cycle graph \( C_{3} \) with a vertex in common. It is denoted by \( F_{3}^{(n)} \) (Kanna et al. 2016). Friendship graph \( F_{3}^{(n)} \) contains \( 2n + 1 \) vertices and \( 3n \) edges as shown in the figures.
As we mentioned, the new nano materials and drugs are pretty useful in developing areas, and the topological indices are helpful to test the chemical properties of them. In this paper, we present the distance based indices of friendship graph \( F_{3}^{(n)} \). The results we obtained here show promising prospects of the application in material and chemical engineering.
Main results
Theorem 1
Also, from Fig. 1 and the edge set of the friendship graph \( F_{3}^{(n)} \), one can see that there are \( 2n \) 1edgepaths between only center vertex and all other vertices with degree 2 and for all two vertices \( v,u \in V(F_{3}^{(n)} ) \) with degree 2, there are \( n \) 1edgepaths. Thus the coefficient of the first term of the Hosoya polynomial of friendship graph \( F_{3}^{(n)} \) is equal to the number of its edges.
For the second term of the Hosoya polynomial of \( F_{3}^{(n)} \), we see that there are \( \frac{(2n)(2n  2)}{2} \) 2edgepaths between all pair of vertices \( v,u \in V(F_{3}^{(n)} ) \) with degree 2. So, the coefficient of the second term of the Hosoya polynomial is equal to \( 2n^{2}  2n \).
Theorem 2
Theorem 3

The Schultz polynomial of \( F_{3}^{(n)} \) is equal to
$$ Sc(F_{3}^{(n)} ,x)\, = \,2n(n + 4)x + + 8n(n  1)x^{2} $$(21) 
The modified Schultz polynomial of \( F_{3}^{(n)} \) is equal to
$$ Sc*(F_{3}^{(n)} ,x)\, = \,4n(n + 1)x + + 8n(n  1)x^{2} $$(22)
The number of all distinct types of 1 and 2edgepaths
iedgepaths  Degrees of vertices uv  Coefficient  Term of Schultz polynomial  Term of modified Schultz polynomial 

1  2n  2n  2n(n+2)  4n ^{ 2 } 
1  22  n  4n  4n 
2  2n  0  0  0 
2  22  2n(n−1)  8n(n−1)  8n(n−1) 
Theorem 4

The Schultz index of the friendship graph \( F_{3}^{(n)} (\forall n \ge 2) \) is equal to
$$ Sc(F_{3}^{(n)} ) = 2n(9n  7) $$(25) 
The modified Schultz index of the friendship graph \( F_{3}^{(n)} (\forall n \ge 2) \) is equal to
$$ Sc*\left( {F_{3}^{(n)} } \right) = \, 4n\left( {5n  3} \right). $$(26)
Conclusions
In this article, by means of graph structure analysis, we have determined the several distancebased topological indices of friendship graph \( F_{3}^{(n)} \) which is widely appeared in various classes of new nanomaterials, drugs and chemical compounds. These results will be helpful to understand the underlying molecular topologies of these graphs.
Declarations
Authors’ contributions
WG and MRF proposed the idea for computing the distancebased topological indices of friendship graph which was implemented and computations were founds by MI and MRRK which were verified by all the authors. The final draft was prepared by WG and MRF. All authors read and approved the final manuscript.
Acknowledgements
This research work is supported by NSFC (Nos. 11401519 and 61262070).
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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