Eccentric topological properties of a graph associated to a finite dimensional vector space

Abstract: A topological index is actually designed by transforming a chemical structure into a number. Topological index is a graph invariant which characterizes the topology of the graph and remains invariant under graph automorphism. Eccentricity based topological indices are of great importance and play a vital role in chemical graph theory. In this article, we consider a graph (nonzero component graph) associated to a finite dimensional vector space over a finite filed in the context of the following eleven eccentricity based topological indices: total eccentricity index; average eccentricity index; eccentric connectivity index; eccentric distance sum index; adjacent distance sum index; connective eccentricity index; geometric arithmetic index; atom bond connectivity index; and three versions of Zagreb indices. Relationship of the investigated indices and their dependency with respect to the involved parameters are also visualized by evaluating them numerically and by plotting their results.


Introduction
Let a connected graph G having the sets V(G) and E(G) as the vertex set and the edge set, respectively. Starting from a vertex v and ending at a vertex u in G, a shortest alternating sequence of vertices and edges without repetition of any vertex is known as a v − u geodesic. The number of edges in a v − u geodesic is denoted by d (v,u), and is called the distance between v and u in G. The maximum distance of a vertex ν among its distances with all the vertices of G is called the eccentricity of v, denoted by ecc (v). The number is the diameter of G. For any vertex v of G, the number is the distance number of v, and the number d(v) is the degree of v. An involvement of different graph theoretic tools/ parameters in the field of chemistry to deal with chemical structures is due to chemical graph theory. In chemical graph theory, an interesting sub field, namely cheminformatics, deals with chemical phenomenal "quantitative structureactivity/structure-property relationships" of chemical compounds. An emerging tool used, in the study of these phenomenal, is a topological index, which is invariant for chemical structures up to their symmetry (automorphism). Correlation of many physico-chemical properties like boiling point, stability, strain energy, etc. of chemical compounds in a chemical structure is due to its topological index (Bruckler et al., 2011;Deng et al., 2011;Klavžar and Gutman, 1996;Liu et al., 2019;Rucker and Rucker, 1999;Tang et al., 2019;Zheng et al., 2019;Zhang and Zhang, 1996). In this regard, the very first topological index was introduced by Wiener in 1947, and gave it name the path number (Wiener, 1947). This index is based on the concept of distance, mathematically, defined as: ( ) and called the Wiener index of a chemical structure/ graph G. Extending the study of topological index, based on the distance, many graph theorists introduced and studied various topological indices. Among these topological indices, a lot of researchers considered topological indices, defined using the eccentricity of each vertex, and produced remarkable investigations, such as: the average eccentricity of a graph and its subgraph was investigated in Dankelmann et al. (2004); the extremal properties of the average eccentricity and conjectures about the average eccentricity were obtained in Ilic (2012); lower and upper bounds of average eccentricity for trees were provided in Tang and Zhou (2012); the average eccentricity and standard deviation of Sierpinski graphs were established in Hinz and Parisse (2012); the eccentricity based ABC and geometric arithmetic indices for copper oxide networks are obtained in Imran et al. (2017); in Gao et al. (2016), results about the eccentric ABC index of linear polyene parallelogram benzenoid are investigated; the total eccentricity, average eccentricity, eccentricitybased Zagreb indices, eccentricity based atom bond connectivity (ABC) and geometric arithmetic indices for oxide networks were found in Imran et al. (2018); eccentricity based topological indices of a cyclic octahedron structure were explored in Zahid et al. (2018). Furthermore, results regarding the average eccentricity index and eccentricity based geometric arithmetic index can be found in Zhang et al. (2017).
In this paper, we extend the study of eccentricity topological indices, based on the eccentricity, in chemical graph theory by involving an algebraic structures called a vector space. We consider a graph associated to a finite dimensional vector space over a finite field, and compute all the indices listed in the . Now, the graph associated with , denoted by Γ( ) , is called a non-zero component graph, which is defined with respect to a basis α α α … { , , , } n 1 2 as follows: the vertex set is θ Γ = − V ( ( )) ; and two vertices form an edge in Γ( ) if v 1 and v 2 share at least one α i with non-zero coefficient in their basic representation, i.e. their exists at least one α i along which both v 1 and v 2 have non-zero components. Unless otherwise mention, we take the basis α α α … { , , , } n 1 2 on which the graph is constructed (Das, 2016). Now, we state some basic results about Γ( ) , which will be useful in the sequel.

Theorem 1 (Das, 2016)
If be an n − dimensional vector space over a field with q ≥ 2 elements, then the order of Γ ( ) is q n − 1 and the size of

Theorem 2 (Das, 2016)
Let be a vector space over a finite field with q ≥ 2 elements and Γ ( ) be its associated graph with respect to basis α α

Theorem 4 (Das, 2016)
Γ ( ) is a complete graph if and only if is 1 − dimensional.
Let be an n − dimensional vector space, n ≥ 1, over a field of order q ≥ 2. Then there are ( ) − ( 1 ) k and 1 k n.
Let us denote the degree of a vertex ( ) For n = 1 and q ≥ 2, Γ( ) is a complete graph by Theorem 4, so one can trivially find all the topological indices in this particular case. We consider Γ( ) for more than 1 − dimensional vector spaces in the context of previously defined eccentricity based topological indices.

Methodology
Some graph theoretical parameters such as path, distance, eccentricity, diameter, degree, etc. along with vertex partitioning method are used to construct some useful tools for investigating our main results. We also use combinatorial computing and binomial expansion theorem to find the required indices. Moreover, we use maple software (Maplesoft, McKinney, TX, USA) (see: https://en.wikipedia.org/wiki/Maplesoftware) for plotting our mathematical results, mathematical calculations and verifications, and to provide a numeric comparison of the investigated indices.

Construction
In this section, we construct/illustrate some useful results to compute the required indices. Let us denote the dimension of a vector space with dim( ) , and the order of a field with o( ) .

Remark 5
2 , then for all 1 ≤ k ≤ n and for each Since the number of vertices
The following proposition provides the distance number of each vertex of Γ( ) .

Proposition 7
Let dim n ( ) 2 = ≥ and o q ( ) 2 = ≥ , then for any 1 ≤ k ≤ n and for each i n k q 1 ( Using Remark 5, the distance number of any vertex of Γ( ) is: The notation a ∼ b is a type of an edge whose end vertices have degrees a and b. The following result gives edge partition of Γ( ) according to the eccentricity of each vertex.

Proposition 8
Let dim n ( ) 2 = ≥ and o q ( ) 2 = ≥ , then the edge partition of Γ( ) according to the eccentricity of each vertex is: where a ∼ b denotes an edge whose one end vertex has the eccentricity a and the other end vertex has the eccentricity b.

Proof
By Remark 6, there are three types of edges of Γ( ) : 1 ∼ 1;1 ∼ 2 and 2 ∼ 2. The number of edges in each type can be found as follows: (1) Edges of type 1 ∼ 1: (2) Edges of type 1 ∼ 2: In Γ( ) , two types of edges, 1 ∼ 1 and 1 ∼ 2, are counted to form a degree d n of a vertex v i n , ≤ ≤ − i q 1 ( 1) n . Thus, the number of edges of type 1 ∼ 2 is obtained from the degree d n of v i n by subtracting the number of edges of type 1 ∼ 1. Note that, − − q ( 1) 1 n edges of type 1 ∼ 1 are counted to form a degree d n , because every two vertices having degree d n are adjacent in Γ( ) . It follows that − − + d q ( 1) 1 n n edges of type 1 ∼ 2 are counted to form each degree d n . Since there are − q ( 1) n vertices having degree d n , so the number of edges of type 1 ∼ 2 is (3) Edges of type 2 ∼ 2: The number of edges of type 2 ∼ 2 can be obtained by the formula Size of Γ( ) − the number of edges of type 1 ∼ 1 − the number of edges of type 2 ∼ 2, which yields that q q q q q q q q 1 2 ( ( 2 1 ) 1 ) 1 2 ( 1) ( ( 1) 1) ( 1) ( ( 1) 1) n n n n n n n n

Results
The total eccentricity index of Γ( ) is computed in the following result:

Theorem 9
Let G ( ) = Γ be a non-zero component graph of a vector space of dimension n ≥ 2 over a finite field of order q ≥ 2. Then G q q ( ) 2( 1) ( 1) .
Using the total eccentricity index, computed in Theorem 9, in the formula of the average eccentricity index, we get the following result:

Theorem 10
Let G ( ) = Γ be a non-zero component graph of a vector space of dimension n ≥ 2 over a finite field of order q ≥ 2.
n n = − − − In the following result, we compute the connective eccentricity index of Γ( )
The eccentric connectivity index of Γ( ) is computed in the next result.

Theorem 12
Let G ( ) = Γ be a non-zero component graph of a vector space of dimension n ≥ 2 over a finite field of order q ≥ 2. Then ( 1) ( 1) (2) ( 1) ( 1)(1) n n n n n n 1 1 ( 1) ( 1) ( 1) n n n n n n ( 1) n n n 2 q q n q q n n q q n n q q ( 1) n n n n nn n n n After some simple calculations, using binomial expansions, we get the required index.
The investigation of the eccentric distance sum index of Γ( ) is given in the following result:

Theorem 13
Let G ( ) = Γ be a non-zero component graph of a vector space of dimension n ≥ 2 over a finite field of order q ≥ 2. Then ( 1) 1 3 n n n n n n n n 1 1 1 ( 1) ( 2) .

Proof
Using the degree, eccentricity and distance number of each vertex, given in Remark 6 and Proposition 7, in the formula of adjacent eccentric distance sum index, we have

Proof
Using the edge partition, given in Proposition 8, in the formula of geometric arithmetic index, we have ( 1) 6 4 2 ( 1) 4 2 6 4 2 6 n n n By performing some calculations, using binomial expansion, we get the required index.
The following result provides the formula for the eccentricity based ABC index of Γ( ) .

Theorem 17
Let G ( ) = Γ be a non-zero component graph of a vector space of dimension n ≥ 2 over a finite field of order q ≥ 2. Then ( 1 ) 2 ( 2 1 ) 2 . n n n n n 1 *

Proof
Using edge partition, given in Proposition 8, in the formula of the first Zagreb eccentricity index, we have ( 1) 1) 1 2 2 n n n n n n 2 q q q q q 1 2 (2 1) ( 1) (2 ( 1) 1) 1 4 n n n n n n 2 Hence, we get the required result.
The following result provides the formulation of the third Zagreb eccentricity index of Γ( ) .

Comparisons and plots
Using maple software (Maplesoft, McKinney, TX, USA), we provide a simple comparison of the investigated indices by plotting them and by constructing tables of their numeric values. We construct Tables 2 and 3 for different values of q,n, which depict that all the indices are in increasing order as the values of q and n are increasing, and their increasing behaviors are clearly shown in Figures 1-11 for certain values of q and n.