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Introduction
In recent years graph theory is extensively used in the branch of mathematical chemistry and some people call it as chemical graph theory because this theory is related with the practical applications of graph theory for solving the molecular problems. In mathematics a model of chemical system portrays a chemical graph that deals to explain the relations between its segments such as its atoms, bonds between atoms, cluster of atoms or molecules.
A connected simple graphG = (V(G) ∪ E(G)) is a graph consisting of n vertices (V(G)) and m edges (E(G)) in which there is path between any of two its vertices. A network is merely a connected graph consisting of no multiple edges and loops. The degree of a vertexv in G is the number of edges which are incident to the vertex v and will be represented by dv. In a graph G, if there is no repetition of vertices in (u − v) walk then such kind of walk is called (u − v) path. The number of edges in (u − v) path is called its length. The distanced(u, v) from vertex u to vertex v is the length of a shortest (u − v) path in a graph G where u, v ∈ G. In a connected graph G, the eccentricityεv of a vertex v is the distance between v and a vertex furthest from v in G. Thus, εv = maxv∈V(G)d(v, u). Therefore the maximum eccentricity over all vertices of G is the diameter of G which is denoted by D(G).
A graph can be recognized by a different type of numeric number, a polynomial, a sequence of numbers or a matrix. A topological index is a numeric quantity that is associated with a graph which characterize the topology of graph and is invariant under graph automorphism. Over the years topological indices like Wiener index Balabans index [24,25,26], Hosoya index [16,17], Randić index [19] and so on, have been studied extensively and recently the research and interest in this area has been increased exponentially. See too for more information [3, 13, 14, 18, 21, 23].
There are some major classes of topological indices such as distance based topological indices, eccentricity based topological indices, degree based topological indices and counting related polynomials and indices of graphs. In this article we shall consider the eccentricity based indices. We note that in [5] is introduced the total eccentricity of a graph G and is defined as the sum of eccentricities of all vertices of a given graph G and denote by ζ(G). It is easy to see that for a k–regular graph G is held ζ(G) = kζ(G).
The Eccentric-connectivity indexξ(G) which was proposed by Sharma, Goswami and Madan defined as [20]:
Another very relevant and special eccentricity based topological index is connective Eccentric indexCξ(G) that was proposed by Gupta et al. in [11]. The connective eccentric index is defined as.
where
$\begin{array}{}
\displaystyle
S_v=\sum\limits_{u\in N(v)}d_u
\end{array}$
that is Sv is the sum of degrees of all vertices adjacent to vertex v.
In 2010, S. Ediz et al., [8], defined Ediz eccentric connectivity index of G as
Similar to other topological polynomials, the corresponding polynomial, that is, the modified eccentric connectivity polynomial of a graph, is defined as, [6]:
where
$\begin{array}{}
\displaystyle
M_v =\prod\limits_{u\in N(v)}d_u
\end{array}$
that is denotes the product of degrees of all neighbors of vertex v of G.
In the same way, we define the modified augmented eccentric connectivity polynomialMAξc(G, x), as
For more information and properties of eccentricity based topological index, see for instance [2, 7, 9, 10, 12, 15, 27].
The aim of this paper is is the introduction of the augmented eccentric connectivity index and modified augmented eccentric connectivity polynomial. As an application we shall compute these new indices for octagonal grid
$\begin{array}{}
\displaystyle
O_n^m
\end{array}$
and we shall compare the results obtained with the ones obtained by other indices like Ediz eccentric connectivity index, modified eccentric connectivity index and modified eccentric connectivity polynomial ECP(G, x) via their computation too.
In [4] and [22] Diudea et al. constructed a C4C8 net as a trivalent decoration made by alternating squares C4 and octagons C8 in two different ways. One is by alternating squares C4 and octagons C8 in different ways denoted by C4C8(S) and other is by alternating rhombus and octagons in different ways denoted by C4C8(R). We denote C4C8(R) by
$\begin{array}{}
\displaystyle
O_n^m
\end{array}$
see Figure 1. In [21] they also called it as the Octagonal grid.
For n, m ≥ 2 the Octagonal grid
$\begin{array}{}
\displaystyle
O_n^m
\end{array}$
, is the grid with m rows and n columns of octagons. The symbols V(
$\begin{array}{}
\displaystyle
O_n^m
\end{array}$
) and E(
$\begin{array}{}
\displaystyle
O_n^m
\end{array}$
) will denote the vertex set and the edge set of
$\begin{array}{}
\displaystyle
O_n^m
\end{array}$
, respectively.
$$\begin{array}{}
\displaystyle
~V (O_n^m ) = \{u_s^t : 1 \leq s \leq n,\ 1\leq t \leq m + 1\}\cup \{ v_s^{t} : 1 \leq s \leq n; \ 1 \leq t \leq m +1 \}\\\displaystyle\qquad\quad~
\cup \{w_s^t : 1 \leq s \leq n+1,\ 1\leq t \leq m \}\cup \{y_s^t : 1 \leq s \leq n+1,\ 1\leq t \leq m \}.
\end{array}$$
$$\begin{array}{}
\displaystyle
~E(O_n^m ) = \{u_s^t v_s^{t} :\, 1
\leq s \leq n, \, 1 \leq t \leq m+1\} \cup \{u_s^{t}w_{s}^{t}; 1 \leq s \leq n,\,
1 \leq t \leq m \} \\\displaystyle
\qquad\quad~\cup \,\{ w_s^{t}y _{s}^{t}: 1 \leq s \leq n+1,\, 1 \leq t \leq m\} \cup \{ v_s^{t}w _{s+1}^{t}: 1 \leq s \leq n,\, 1 \leq t \leq m\} \\\displaystyle
\qquad\quad~\cup \, \{ v_{s}^t y_{s+1}^{t-1}: 1 \leq s \leq n,\, 2\leq t\leq m+1\}\cup \{u_s^{t+1}
y_{s}^{t}:\, 1 \leq s \leq n, \, 2 \leq t \leq m\}.
\end{array}$$
In this paper, we consider
$\begin{array}{}
\displaystyle
O_n^m
\end{array}$
with n = m.
Statement of main results
As we have said previously for
$\begin{array}{}
\displaystyle
O_n^m
\end{array}$
with n = mwe shall compute modified eccentric connective index, Ediz eccentric connectivity index, modified eccentric connective polynomial, modified augmented eccentric connective index and modified augmented eccentric connectivepolynomial and we shall compare the results obtained. For this we have discussed two cases of n, when n ≡ 0(mod 2) and when n ≡ 1(mod 2). Also to avoid any ambiguity related to Figure 1 note that the vertices
$\begin{array}{}
\displaystyle
u_s^t=\text{u}_s^t
\end{array}$.
Theorem 1
For every n ≥ 4 and n ≡ 0 (mod 2) consider the graph of G ≅
$\begin{array}{}
\displaystyle
O_n^m
\end{array}$
, with n = m. Then the modified eccentric connectivity index ξc(G) of G is equal to
Let G be the graph of
$\begin{array}{}
\displaystyle
O_n^m
\end{array}$
. Note that graph of
$\begin{array}{}
\displaystyle
O_n^m
\end{array}$
is a symmetric about reflection and rotation at right angles. Thus the eccentricities
$\begin{array}{}
\displaystyle
\varepsilon_{u_s^t}=\varepsilon_{v_{n+1-s}^{t}}
\end{array}$
and from the symmetry at right angles we can obtain that the eccentricities
$\begin{array}{}
\displaystyle
\varepsilon_{y_s^t}=\varepsilon_{u_t^s},~\varepsilon_{w_s^t}=\varepsilon_{v_t^s}
\end{array}$
. Therefore, from Table 1 and formula (3), given below, the modified eccentric connectivity index ξc(G) of
$\begin{array}{}
\displaystyle
O_n^m
\end{array}$
is equal to
Partition of vertices of the type
$\begin{array}{}
\displaystyle
u_s^t~~ \text{of}~~ O_n^m
\end{array}$
based on degree sum and eccentricity of each vertex when n ≡ 0( mod 2).
For every n ≥ 3 and n ≡ 1 (mod 2) consider the
graph of G ≅
$\begin{array}{}
\displaystyle
O_n^m
\end{array}$
, with n = m. Then the modified eccentric connectivity index ξc(G) of G is equal to
Let G be the graph of
$\begin{array}{}
\displaystyle
O_n^m
\end{array}$
and n ≥ 3 is odd. As above note that graph of
$\begin{array}{}
\displaystyle
O_n^m
\end{array}$
is a symmetric about reflection and rotation at right angles. Thus the eccentricities
$\begin{array}{}
\displaystyle
\varepsilon_{u_s^t}=\varepsilon_{v_{n+1-s}^{t}}
\end{array}$
and from the symmetry at right angles we can obtain that the eccentricities
$\begin{array}{}
\displaystyle
\varepsilon_{y_s^t}=\varepsilon_{u_t^s},~\varepsilon_{w_s^t}=\varepsilon_{v_t^s}
\end{array}$
. Therefore, by using Table 2 and equation (3) the modified eccentric connectivity index ξc(G), we get
Partition of vertices of the type
$\begin{array}{}
\displaystyle
u_s^t~~ \text{of}~~ O_n^m
\end{array}$
based on degree sum and eccentricity of each vertex when n ≡ 1(mod 2).
For every n ≥ 4 and n ≡ 0 (mod 2) consider the graph of G ≅
$\begin{array}{}
\displaystyle
O_n^m
\end{array}$
, with n = m. Then the Ediz eccentric connectivity index of G is equal to
Let G be the graph of
$\begin{array}{}
\displaystyle
O_n^m
\end{array}$
and n ≡ 0 (mod2 ). By using the arguments in proof of Theorem 1, Table 1 and following formula the Ediz eccentric connectivity index Eξc(G) of
$\begin{array}{}
\displaystyle
O_n^m
\end{array}$
is equal to
For everyn ≥ 3 andn ≡ 1 ( mod 2) consider the graph ofG ≅
$\begin{array}{}
\displaystyle
O_n^m
\end{array}$
, withn = m. Then the Ediz eccentric connectivity index ofGis equal to
Let G be the graph of
$\begin{array}{}
\displaystyle
O_n^m
\end{array}$
and n ≡ 1 ( mod 2). By using the arguments in proof the of Theorem 2, Table 2 and from formula (4) the result follows. □
Theorem 5
For every n ≥ 4 and n ≡ 0 ( mod 2) consider the graph of G ≅
$\begin{array}{}
\displaystyle
O_n^m
\end{array}$
, with n = m. Then the modified eccentric connectivity polynomial of G is equal to
For everyn ≥ 3 andn ≡ 1 ( mod 2) consider the graph ofG ≅
$\begin{array}{}
\displaystyle
O_n^m
\end{array}$
, withn = m. Then the modified eccentric connectivity polynomial ofGis equal to
Let G ≅
$\begin{array}{}
\displaystyle
O_n^m
\end{array}$
, n ≥ 3 and n ≡ 1 ( mod 2). By using the arguments in the proof of Theorem 1, as in Theorem 5, the values from Table 2 and equation (5) the result follows. □
In Table 3 and Table 4 we have partitioned the vertices of the type
$\begin{array}{}
\displaystyle
u_s^t~~ \text{of}~~ O_n^m
\end{array}$
based on degree product and eccentricity of each vertex. This will help us to develop the coming theorems.
Partition of vertices of the type
$\begin{array}{}
\displaystyle
u_s^t~~ \text{of}~~ O_n^m
\end{array}$
based on degree product and eccentricity of each vertex when n ≡ 0( mod 2).
Partition of vertices of the type
$\begin{array}{}
\displaystyle
u_s^t~~ \text{of}~~ O_n^m
\end{array}$
based on degree product and eccentricity of each vertex when n ≡ 1( mod 2).
For every n ≥ 4 and n ≡ 0 ( mod 2) consider the graph of G ≅
$\begin{array}{}
\displaystyle
O_n^m
\end{array}$
, with n = m. Then the modified augmented eccentric connectivity index
$\begin{array}{}
\displaystyle
^{MA}\xi^c_c(G)
\end{array}$
of G is equal to
Let G be the graph of
$\begin{array}{}
\displaystyle
O_n^m
\end{array}$
. Therefore, from Table 3 and formula (8), given below, the modified augmented eccentric connectivity index
$\begin{array}{}
\displaystyle
^{MA}\xi^c_c(O_n^m)~~ \text
{of}~~ O_n^m
\end{array}$
can be calculated. Hence the result.
For every n ≥ 3 and n ≡ 1( mod 2) consider the
graph of G ≅
$\begin{array}{}
\displaystyle
O_n^m
\end{array}$
, with n = m. Then the modified augmented eccentric connectivity index ξc(G) of G is equal to
For every n ≥ 4 and n ≡ 0 ( mod 2) consider the graph of G ≅
$\begin{array}{}
\displaystyle
O_n^m
\end{array}$
, with n = m. Then the modified augmented eccentric connectivity polynomial of G is equal to
For everyn ≥ 3 andn ≡ 1 ( mod 2) consider the graph ofG ≅
$\begin{array}{}
\displaystyle
O_n^m
\end{array}$
, withn = m. Then the modified augmented eccentric connectivity polynomial ofGis equal to
As in Theorem 9, by using Table 4 and the equation (8) the result follows. □
Conclusions and comparison between the indices
High discriminating power and extremely low degeneracy are desirable properties of an ideal topological index, which researchers in theoretical chemistry are striving to achieve. The values of MAξc(G) were computed for all the possible structure of three and four vertices. The values and the structures have been presented in Table 5 and their comparison is presented in Table 6. Modified augmented eccentric connectivity index demonstrate exceptionally high discriminating power, defined as the ratio of the highest to lowest value for all possible structures with the same number of vertices. This is evident from the fact that the ratio of the highest to lowest value for all possible structure containing three and four vertices is very high in contrast to ξ(G) and ξc(G). The ratio of the highest to lowest value for all possible structures containing four vertices for MAξc(G) is 6.75 in comparison to 1.78 and 1.7 for ξ(G) and ξc(G), respectively. The exceptionally high discriminating power of the proposed indices makes them extremely sensitive towards minor change(s) in molecular structure. This extreme sensitivity towards branching and the discriminating power of proposed indices are clearly evident from the respective index values of all the possible structures with four vertices.
Values of eccentric connectivity index, modified eccentric connectivity index and modified augmented eccentric connectivity index for all possible structures with three and four.
S.N
Structure
ξ(G)
MAξc(G)
ξc(G)
1
6
9
10
2
6
12
12
3
14
16
24
4
9
19
21
5
13
32
32
6
16
32
32
7
14
60
29
8
12
108
36
Comparison of the discriminating power and degeneracy of eccentric connectivity index, modified eccentric connectivity index and modified augmented eccentric connectivity index using all possible structures with three and four vertices.
ξ(G)
MAξc(G)
ξc(G)
• For three vertices
Minimum value
6
9
10
Maximum value
6
12
12
Ratio
1:1
1:1.34
1:1.2
Degeneracy
1/2
0/2
0/2
• For four vertices
Minimum value
9
16
21
Maximum value
16
108
36
Ratio
1:1.78
1:6.75
1:1.7
Degeneracy
1/6
1/6
1/6
Degeneracy: the number of compounds having identical values/the total number of compounds with the same number of vertices.
Degeneracy is a measure of the ability of an index to differentiate between the relative positions of atom in a molecule. MAξc(G) did not exhibit any degeneracy for all possible structures with three vertices whereas MAξc(G) had a very low degeneracy of one in the case of all possible structures with four vertices (Table 6). ξ(G) had one identical values out of 6 structures with only four vertices. Extremely low degeneracy indicates the enhanced capability of these indices to differentiate and demonstrate slight variations in the molecular structure, which clearly reveals the remote chance of different structures having the same value.
The Table 7 shows a comparison between the eccentric connectivity index, modified eccentric connectivity index and modified augmented eccentric connectivity index for octagonal grid
$\begin{array}{}
\displaystyle
O_n^m
\end{array}$
for finite n = 3, …, 10.
comparison of
$\begin{array}{}
\displaystyle
\xi_c(O_m^n), ~ ^E\xi^c(O_m^n)~\text{and}~ ^{MA}\xi^c(O_m^n)~\text{for}~ O_m^n,
\end{array}$
when m = n.