Eccentricity-Based Topological Invariants of Some Chemical Graphs

atoms Article Eccentricity-Based Topological Invariants of Some Chemical Graphs Nazeran Idrees 1, * , Muhammad Jawwad Saif 2 and Tehmina Anwar 1 1 2 * Department of Mathematics, Government College University Faisalabad, Punjab 38000, Pakistan; Department of Applied Chemistry, Government College University Faisalabad, Punjab 38000, Pakistan; Correspondence: Received: 25 November 2018; Accepted: 24 January 2019; Published: 6 February 2019



 Abstract: Topological index is an invariant of molecular graphs which correlates the structure with different physical and chemical invariants of the compound like boiling point, chemical reactivity, stability, Kovat’s constant etc. Eccentricity-based topological indices, like eccentric connectivity index, connective eccentric index, first Zagreb eccentricity index, and second Zagreb eccentricity index were analyzed and computed for families of Dutch windmill graphs and circulant graphs. Keywords: chemical graph; topological index; eccentricity; circulant graph; Dutch windmill graph 1. Introduction A single number which represents a chemical structure, in graph-theoretical chemistry, is called a topological descriptor (or index). A topological index is a real number which correlates the structure of chemical compound with their chemical reactivity or physical properties. Chemical graph theory is well-known branch of graph theory which concerns with mathematical modeling of molecules. It also deals with the study of development of topological indices, isomerism, and found applications in quantum chemistry and stereochemistry. Topological indices are mainly used in quantitative structure–activity relations (QSAR) as well as quantitative structure–property relations (QSPR) which describe the relation between chemical structure with the properties and reactivity of the compounds. Chemical structure is depicted as graphs with vertices representing atoms and the edges represent the chemical bonds between atoms. Let G (V, E) be a simple and connected graph with n vertices and m edges. Let u ∈ V then be the eccentricity of a vertex where u is a maximum distance of u from other vertices of graph G, which is denoted by ε(u), i.e., ε(u) = max{d(u, v); v ∈ V }, where d(u, v) is a distance between u and v. The degree of a vertex u, denoted by d(u), is number of vertices which are attached to u by the edges. The eccentric connectivity index is introduced by Sharma, Goswami, and Madan [1], and defined as ξ c = ∑ d ( u ) ε ( u ). (1) u ∈V In 2000, Gupta, Singh, and Madan [2] introduced a topological descriptor termed the connective eccentricity index, which is defined as d(u) Cξ = ∑ . (2) ε u u ∈V ( ) Atoms 2019, 7, 21; doi:10.3390/atoms7010021 www.mdpi.com/journal/atoms Atoms 2019, 7, 21 2 of 9 The Zagreb indices were introduced more than thirty years ago by Gutman and Trinajestic [3]. They are defined as M1∗ = ∑ d2 (u), u ∈V M2∗ = ∑ d(u)d(v). uv∈ E After thirty years, new version of Zagreb indices introduced by Ghorbani and Hosseinzadeh [4] are first and second Zagreb eccentricity index, which are stated as M1 ( G ) = ∑ ε2 ( z ), (3) ∑ ε ( y ) ε ( z ). (4) z ∈V ( G ) M2 ( G ) = yz∈ E( G ) Khalifeh et al. [5] calculated the Zagreb indices of arbitrary C4 tube, C4 torus, and q-multiwalled polyhex nanotorus. Doslic et al. [6] gave formulae of the eccentric connectivity index for armchair hexagonal belts, zigzag belts, and the corresponding open chains. Ashrafi et al. [7] found formulas for the eccentric connectivity index of TUC4 C8 (S) nanotube and TC4 C8 (S) nanotorus. Ghorbani [8] derived bounds of the connective eccentric index and calculated connective eccentricity index for two classes of fullerenes which are infinite. Ilić [9] presented the unicyclic graphs and extremal trees with minimum and maximum eccentric connectivity index subject to the certain graph constraints. Ilić et al. [10] derived explicit formulae for the eccentric distance sum for the Cartesian product and joining of graphs. Morgan et al. [11] showed a quite low lower bound for a tree on the eccentric connectivity index, in expressions of diameter and order. Songhori [12] computed the eccentric connectivity for an infinite class of fullerene graphs. Recently, Gao et al. provided several interesting results about topological indices and their applications in biological sciences [13] and nanoscience [14], which are quite promising and motivating for further studies in the area. 2. Results and Discussions In this section, we will compute the connective eccentricity, eccentric connective, first Zagreb eccentricity index, and second Zagreb eccentricity indices of Dutch windmill graph and circulant graph by analyzing the eccentricities of the vertices of the graphs. n obtained by joining n numbers of cycle graphs C with a Dutch Windmill graph: A graph Dm m common vertex is known as Dutch windmill graph. The Dutch windmill graph is an undirected and planar graph.   Circulant graph: Let a1 , a2 , a3 , . . . , am be positive integers where 1 ≤ ai ≤ n2 , ai 6= a j ∀ 1 ≤ i, j ≤ m, and i 6=n j. An undirected and simpleo graph with vertex V = {u1 , u2 , u3 , . . . , un }, and the edge set is E = ui ui+ a j ; 1 ≤ i ≤ n, 1 ≤ j ≤ m which is called the circulant graph and is denoted by Cn ( a1 , a2 , a3 , . . . , am ). The indices being taken modulo n. The numbers a1 , a2 , a3 , . . . , am are called generators. A circulant graph is a regular graph. Let r denote the degree of vertices of the graph, then ( r= 2m − 1, 2m i f n2 ∈ { a1 , a2 , a3 , . . . , am } . otherwise These graphs correspond to wide variety of chemical graphs. For instance, the Dutch windmill graph represents bidentate ligands, as can be seen in the Figure 1 below. Atoms 2019, 7, 21 3 of 9 Figure 1. Tris(ethylenediamine)cobalt(III) chloride and tris(ethylenediamine)chromium(III) sulfate. n ), is given by Theorem 1. The connective eccentricity index of Dutch windmill graph, denoted by C ξ ( Dm n C ξ ( Dm )=  bmc 1 2n   b m c + 4n ∑ j=21 (b m c+ j) ; 2   6n m + 4n m −2 ∑ j=21 ( i f m is odd 2 1 i f m is even ; m +j 2 , ) n ), is given by and the eccentric connective index, denoted by ξ c ( Dm  m  
 2nb m c + 4n ∑b 2 c m + j ; 2 2 j =1 c n ξ ( Dm ) = m −2  3mn + 4n ∑ 2 m + j
; j =1 2 i f m is odd . i f m is even n be the Dutch windmill graph with n copies of cycle C having common vertex z with Proof. Let Dm m degree (z) = 2n. The degree of other vertices of the graph is two.    Case 1: m is odd. The eccentricity of the central vertex is ε(z) = m2 and eccentricity of other vertices increase by one as we move away from the common vertex to the half of the cycle, as can be seen in Figure 2. When m odd, the vertices other than the common vertex are even in number in each cycle. The eccentricity of the vertices, in each cycle, is pairwise equal, and are equidistant from n have a total 2n vertices of same eccentricity, which are the central (...truncated)