New and Modified Eccentric Indices of Octagonal Grid

Representative	$\begin{matrix} S_{u_{s}^{t}} \end{matrix}$ $\begin{array}{} \displaystyle S_{u_s^t} \end{array}$	eccentricity	Range	Frequency
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	4	4n − s + 1	t = 1, n + 1; s = 1	2
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	5	4n − s + 1	t = 1, n + 1,	n − 1
			2 ≤ s ≤ $\begin{matrix} \frac{n + 1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	5	3n + s − 1	t = 1, n + 1,	n − 1
			$\begin{matrix} \frac{n + 1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ + 1 ≤ s ≤ n
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	7	4n − 3(s − 1) − t	1 = s,	( $\begin{matrix} \frac{n + 1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ − 1)
			2 ≤ t ≤ $\begin{matrix} \frac{n + 1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	9	4n − 3(s − 1) − t	2 ≤ s ≤ $\begin{matrix} \frac{n + 1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ − 1,	$\begin{matrix} \frac{n - 3}{4} (\frac{n + 1}{2} - 1) \end{matrix}$ $\begin{array}{} \displaystyle \frac{ n-3}{4}(\,\,\, \frac{n+1}{2}-1\,\,) \end{array}$
			s + 1 ≤ t ≤ $\begin{matrix} \frac{n + 1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	9	4(n + 1) − s − 3t	2 ≤ s ≤ $\begin{matrix} \frac{n + 1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$,	$\begin{matrix} \frac{n + 1}{4} (\frac{n + 1}{2} - 1) \end{matrix}$ $\begin{array}{} \displaystyle \frac{ n+1}{4}(\,\,\, \frac{ n+1}{2}-1\,\,) \end{array}$
			2 ≤ t ≤ s
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	9	3n + s − 3t + 2	$\begin{matrix} \frac{n + 1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ + 1 ≤ s ≤ n − 1,	$\begin{matrix} \frac{n - 1}{4} (\frac{n - 1}{2} - 1) \end{matrix}$ $\begin{array}{} \displaystyle \frac{ n-1}{4}(\,\,\, \frac{ n-1}{2}-1\,\,) \end{array}$
			2 ≤ t ≤ n + 1 − s
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	9	n + 3s − t − 1	$\begin{matrix} \frac{n + 1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ + 1 ≤ s ≤ n,	$\begin{matrix} \frac{n - 1}{4} (\frac{n - 1}{2} + 1) \end{matrix}$ $\begin{array}{} \displaystyle \frac{n-1}{4}(\,\,\, \frac{n-1}{2}+1\,\,) \end{array}$
			n − s + 2 ≤ t ≤ $\begin{matrix} \frac{n + 1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	7	3(n − s) + t + 1	s = 1,	( $\begin{matrix} \frac{n + 1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ − 1)
			$\begin{matrix} \frac{n + 1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ + 1 ≤ t ≤ n
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	9	n − s + 3t − 2	2 ≤ s ≤ $\begin{matrix} \frac{n + 1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$,	$\begin{matrix} \frac{n - 3}{4} (\frac{n + 1}{2} - 1) \end{matrix}$ $\begin{array}{} \displaystyle \frac{ n-3}{4}(\,\,\, \frac{n+1}{2}-1\,\,) \end{array}$
			n − s + 2 ≤ t ≤ n
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	9	4s − n + t − 3	$\begin{matrix} \frac{n + 1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ + 1 ≤ s ≤ n,	$\begin{matrix} \frac{n - 1}{4} (\frac{n + 1}{2}) \end{matrix}$ $\begin{array}{} \displaystyle \frac{ n-1}{4}(\,\,\, \frac{ n+1}{2}\,\,) \end{array}$
			$\begin{matrix} \frac{n + 1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ + 1 ≤ t ≤ s
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	9	s + 3t − 4	$\begin{matrix} \frac{n + 1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ + 1 ≤ s ≤ n − 1,	$\begin{matrix} \frac{n - 1}{4} (\frac{n - 1}{2} - 1) \end{matrix}$ $\begin{array}{} \displaystyle \frac{ n-1}{4}(\,\,\, \frac{ n-1}{2}-1\,\,) \end{array}$
			s + 1 ≤ t ≤ n

Partition of vertices of the type ust of Onm$\begin{array}{} \displaystyle u.s^t \text{of} O.n^m \end{array}$ based on degree product and eccentricity of each vertex when n ≡ 1( mod 2).

Representative	$\begin{matrix} M_{u_{s}^{t}} \end{matrix}$ $\begin{array}{} \displaystyle M_{u_s^t} \end{array}$	eccentricity	Range	Frequency
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	4	4n − s + 1	t = 1, n + 1; s = 1	2
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	6	4n − s + 1	t = 1, n + 1,	n − 1
			2 ≤ s ≤ $\begin{matrix} \frac{n + 1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	6	3n + s − 1	t = 1, n + 1,	n − 1
			$\begin{matrix} \frac{n + 1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ + 1 ≤ s ≤ n
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	12	4n − 3(s − 1) − t	1 = s,	( $\begin{matrix} \frac{n + 1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ − 1)
			2 ≤ t ≤ $\begin{matrix} \frac{n + 1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	27	4n − 3(s − 1) − t	2 ≤ s ≤ $\begin{matrix} \frac{n + 1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ − 1,	$\begin{matrix} \frac{n - 3}{4} (\frac{n + 1}{2} - 1) \end{matrix}$ $\begin{array}{} \displaystyle \frac{ n-3}{4}(\,\,\, \frac{n+1}{2}-1\,\,) \end{array}$
			s + 1 ≤ t ≤ $\begin{matrix} \frac{n + 1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	27	4(n + 1) − s − 3t	2 ≤ s ≤ $\begin{matrix} \frac{n + 1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$,	$\begin{matrix} \frac{n + 1}{4} (\frac{n + 1}{2} - 1) \end{matrix}$ $\begin{array}{} \displaystyle \frac{ n+1}{4}(\,\,\, \frac{ n+1}{2}-1\,\,) \end{array}$
			2 ≤ t ≤ s
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	27	3n + s − 3t + 2	$\begin{matrix} \frac{n + 1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ + 1 ≤ s ≤ n − 1,	$\begin{matrix} \frac{n - 1}{4} (\frac{n - 1}{2} - 1) \end{matrix}$ $\begin{array}{} \displaystyle \frac{ n-1}{4}(\,\,\, \frac{ n-1}{2}-1\,\,) \end{array}$
			2 ≤ t ≤ n + 1 − s
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	27	n + 3s − t − 1	$\begin{matrix} \frac{n + 1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ + 1 ≤ s ≤ n,	$\begin{matrix} \frac{n - 1}{4} (\frac{n - 1}{2} + 1) \end{matrix}$ $\begin{array}{} \displaystyle \frac{n-1}{4}(\,\,\, \frac{n-1}{2}+1\,\,) \end{array}$
			n − s + 2 ≤ t ≤ $\begin{matrix} \frac{n + 1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	12	3(n − s) + t + 1	s = 1,	( $\begin{matrix} \frac{n + 1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ − 1)
			$\begin{matrix} \frac{n + 1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ + 1 ≤ t ≤ n
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	27	n − s + 3t − 2	2 ≤ s ≤ $\begin{matrix} \frac{n + 1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$,	$\begin{matrix} \frac{n - 3}{4} (\frac{n + 1}{2} - 1) \end{matrix}$ $\begin{array}{} \displaystyle \frac{ n-3}{4}(\,\,\, \frac{n+1}{2}-1\,\,) \end{array}$
			n − s + 2 ≤ t ≤ n
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	27	4s − n + t − 3	$\begin{matrix} \frac{n + 1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ + 1 ≤ s ≤ n,	$\begin{matrix} \frac{n - 1}{4} (\frac{n + 1}{2}) \end{matrix}$ $\begin{array}{} \displaystyle \frac{ n-1}{4}(\,\,\, \frac{ n+1}{2}\,\,) \end{array}$
			$\begin{matrix} \frac{n + 1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ + 1 ≤ t ≤ s
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	27	s + 3t − 4	$\begin{matrix} \frac{n + 1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ + 1 ≤ s ≤ n − 1,	$\begin{matrix} \frac{n - 1}{4} (\frac{n - 1}{2} - 1) \end{matrix}$ $\begin{array}{} \displaystyle \frac{ n-1}{4}(\,\,\, \frac{ n-1}{2}-1\,\,) \end{array}$
			s + 1 ≤ t ≤ n

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)	^MAξ^c(G)	ξ_c(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

Partition of vertices of the type ust of Onm$\begin{array}{} \displaystyle u.s^t \text{of} O.n^m \end{array}$ based on degree sum and eccentricity of each vertex when n ≡ 0( mod 2).

Representative	$\begin{matrix} S_{u_{s}^{t}} \end{matrix}$ $\begin{array}{} \displaystyle S_{u_s^t} \end{array}$	eccentricity	Range	Frequency
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	4	4n - s + 1	t = 1, n + 1; s = 1	2
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	5	4n − s + 1	t = 1, n + 1,	n
			2 ≤ s ≤ $\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 1
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	5	3n + s − 1	t = 1, n + 1,	n − 2
			$\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 2 ≤ s ≤ n
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	7	4n − 3(s − 1) − t	s = 1,	$\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$
			2 ≤ t ≤ $\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 1
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	9	4n − 3(s − 1) − t	2 ≤ s ≤ $\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$,	$\begin{matrix} \frac{n^{2}}{8} - \frac{3 n}{4} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n^2}{8}-\frac{3n}{4} \end{array}$
			s + 1 ≤ t ≤ $\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 1
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	9	4(n + 1) − s − 3t	2 ≤ s ≤ $\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$,	$\begin{matrix} \frac{n^{2}}{8} - \frac{n}{4} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n^2}{8}- \frac{ n}{4} \end{array}$
			2 ≤ t ≤ s
$\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$	9	3n + s − 3t + 2	$\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 1 ≤ s ≤ n − 1,	$\begin{matrix} \frac{n^{2}}{8} - \frac{n}{4} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n^2}{8}- \frac{ n}{4} \end{array}$
			2 ≤ t ≤ n + 1 − s
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	9	n + 3s − t − 1	$\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 1 ≤ s ≤ n,	$\begin{matrix} \frac{n^{2}}{8} - \frac{n}{4} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n^2}{8}- \frac{ n}{4} \end{array}$
			n − s + 2 ≤ t ≤ $\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 1
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	7	3(n − s) + t + 1	s = 1,	$\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ − 1
			$\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 2 ≤ t ≤ n
$\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$	9	3(n − s) + t + 1	2 ≤ s ≤ $\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ − 1,	$\begin{matrix} \frac{1}{8} \end{matrix}$ $\begin{array}{} \displaystyle \frac{1}{8} \end{array}$(n − 4)(n − 2)
			$\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 2 ≤ t ≤ n − s + 1
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	9	n − s + 3t − 2	2 ≤ s ≤ $\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$,	$\begin{matrix} \frac{n^{2}}{8} - \frac{n}{4} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n^2}{8}- \frac{ n}{4} \end{array}$
			n − s + 2 ≤ t ≤ n
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	9	4s − n + t − 3	$\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 2 ≤ s ≤ n,	$\begin{matrix} \frac{n^{2}}{8} - \frac{n}{4} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n^2}{8}- \frac{ n}{4} \end{array}$
			$\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 2 ≤ t ≤ s
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	9	s + 3t − 4	$\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 1 ≤ s ≤ n − 1,	$\begin{matrix} \frac{n^{2}}{8} - \frac{n}{4} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n^2}{8}- \frac{ n}{4} \end{array}$
			s + 1 ≤ t ≤ n

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	ξ(G)	^MAξ^c(G)	ξ_c(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 ξc(Omn), Eξc(Omn) and MAξc(Omn) for Omn,$\begin{array}{} \displaystyle \xi.c(O.m^n), ~ ^E\xi^c(O.m^n)~\text{and}~ ^{MA}\xi^c(O.m^n)~\text{for}~ O.m^n, \end{array}$ when m = n.

[n, m]	ξ_c( $\begin{matrix} O_{n}^{m} \end{matrix}$ $\begin{array}{} \displaystyle O_n^m \end{array}$)	^Eξ^c( $\begin{matrix} O_{n}^{m} \end{matrix}$ $\begin{array}{} \displaystyle O_n^m \end{array}$)	^MAξ^c( $\begin{matrix} O_{n}^{m} \end{matrix}$ $\begin{array}{} \displaystyle O_n^m \end{array}$)
[3, 3]	2888	$\begin{matrix} \frac{5369}{165} \end{matrix}$ $\begin{array}{} \displaystyle \cfrac{5369}{165} \end{array}$	5880
[4, 4]	6564	$\begin{matrix} \frac{616039}{15015} \end{matrix}$ $\begin{array}{} \displaystyle \cfrac{616039}{15015} \end{array}$	14456
[5, 5]	13460	$\begin{matrix} \frac{74609462}{1322685} \end{matrix}$ $\begin{array}{} \displaystyle \cfrac{74609462}{1322685} \end{array}$	32260
[6, 6]	22972	$\begin{matrix} \frac{563513878}{8580495} \end{matrix}$ $\begin{array}{} \displaystyle \cfrac{563513878}{8580495} \end{array}$	56868
[7, 7]	37280	$\begin{matrix} \frac{803471620793}{10039179150} \end{matrix}$ $\begin{array}{} \displaystyle \cfrac{803471620793}{10039179150} \end{array}$	95576
[8, 8]	55364	$\begin{matrix} \frac{1361165885969}{15168440430} \end{matrix}$ $\begin{array}{} \displaystyle \cfrac{1361165885969}{15168440430} \end{array}$	144424
[9, 9]	79784	$\begin{matrix} \frac{1929246726361}{18627909300} \end{matrix}$ $\begin{array}{} \displaystyle \cfrac{1929246726361}{18627909300} \end{array}$	212136
[10, 10]	109176	$\begin{matrix} \frac{1149037176620287}{10119188365650} \end{matrix}$ $\begin{array}{} \displaystyle \cfrac{1149037176620287}{10119188365650} \end{array}$	293432

Partition of vertices of the type ust of Onm$\begin{array}{} \displaystyle u.s^t \text{of} O.n^m \end{array}$ based on degree product and eccentricity of each vertex when n ≡ 0( mod 2).

Representative	$\begin{matrix} M_{u_{s}^{t}} \end{matrix}$ $\begin{array}{} \displaystyle M_{u_s^t} \end{array}$	eccentricity	Range	Frequency
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	4	4n − s + 1	t = 1, n + 1; s = 1	2
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	6	4n − s + 1	t = 1, n + 1,	n
			2 ≤ s ≤ $\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 1
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	6	3n + s − 1	t = 1, n + 1,	n − 2
			$\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 2 ≤ s ≤ n
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	12	4n − 3(s − 1) − t	s = 1,	$\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$
			2 ≤ t ≤ $\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 1
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	27	4n − 3(s − 1) − t	2 ≤ s ≤ $\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$,	$\begin{matrix} \frac{n^{2}}{8} - \frac{3 n}{4} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n^2}{8}-\frac{3n}{4} \end{array}$
			s + 1 ≤ t ≤ $\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$ + 1
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	27	4(n + 1) − s − 3t	2 ≤ s ≤ $\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$,	$\begin{matrix} \frac{n}{4} (\frac{n}{2} - 1) \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{4}(\,\,\, \frac{ n}{2}-1\,\,) \end{array}$
			2 ≤ t ≤ s
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	27	3n + s − 3t + 2	$\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 1 ≤ s ≤ n − 1,	$\begin{matrix} \frac{n}{4} (\frac{n}{2} - 1) \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{4}(\,\,\, \frac{ n}{2}-1\,\,) \end{array}$
			2 ≤ t ≤ n + 1 − s
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	27	n + 3s − t − 1	$\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 1 ≤ s ≤ n,	$\begin{matrix} \frac{n}{4} (\frac{n}{2} + 1) \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{4}(\,\,\, \frac{ n}{2}+1\,\,) \end{array}$
			n − s + 2 ≤ t ≤ $\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 1
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	12	3(n − s) + t + 1	s = 1,	$\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ − 1
			$\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 2 ≤ t ≤ n
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	27	3(n − s) + t + 1	2 ≤ s ≤ $\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ − 1,	$\begin{matrix} \frac{1}{8} \end{matrix}$ $\begin{array}{} \displaystyle \frac{1}{8} \end{array}$(n − 4)(n − 2)
			$\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 2 ≤ t ≤ n-s + 1
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	27	n − s + 3t − 2	2 ≤ s ≤ $\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$,	$\begin{matrix} \frac{n}{4} (\frac{n}{2} - 1) \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{4}(\,\,\, \frac{ n}{2}-1\,\,) \end{array}$
			n − s + 2 ≤ t ≤ n
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	27	4s − n + t − 3	$\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 2 ≤ s ≤ n,	$\begin{matrix} \frac{n}{4} (\frac{n}{2} - 1) \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{4}(\,\,\, \frac{ n}{2}-1\,\,) \end{array}$
			$\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 2 ≤ t ≤ s
$\begin{matrix} u_{s}^{t} \end{matrix}$ $\begin{array}{} \displaystyle u_s^t \end{array}$	27	s + 3t − 4	$\begin{matrix} \frac{n}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 1 ≤ s ≤ n − 1,	$\begin{matrix} \frac{n}{4} (\frac{n}{2} - 1) \end{matrix}$ $\begin{array}{} \displaystyle \frac{n}{4}(\,\,\, \frac{ n}{2}-1\,\,) \end{array}$
			s + 1 ≤ t ≤ n

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New and Modified Eccentric Indices of Octagonal Grid O^m_n

Published Online: Oct 03, 2018

Page range: 209 - 228

Received: Apr 19, 2018

Accepted: May 26, 2018

DOI: https://doi.org/10.21042/AMNS.2018.1.00016

Keywords
Degree, Eccentricity, Ediz eccentric connectivity index, modified eccentric connectivity index, modified eccentric connectivity polynomial (, ), octagonal grid

© 2018 M. Naeem et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Fig. 1

Partition of vertices of the type ust of Onm$\begin{array}{} \displaystyle u.s^t \text{of} O.n^m \end{array}$ based on degree sum and eccentricity of each vertex when n ≡ 1(mod 2).

Partition of vertices of the type ust of Onm$\begin{array}{} \displaystyle u.s^t \text{of} O.n^m \end{array}$ based on degree product and eccentricity of each vertex when n ≡ 1( mod 2).

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.

Partition of vertices of the type ust of Onm$\begin{array}{} \displaystyle u.s^t \text{of} O.n^m \end{array}$ based on degree sum and eccentricity of each vertex when n ≡ 0( mod 2).

Values of eccentric connectivity index, modified eccentric connectivity index and modified augmented eccentric connectivity index for all possible structures with three and four.

comparison of ξc(Omn), Eξc(Omn) and MAξc(Omn) for Omn,$\begin{array}{} \displaystyle \xi.c(O.m^n), ~ ^E\xi^c(O.m^n)~\text{and}~ ^{MA}\xi^c(O.m^n)~\text{for}~ O.m^n, \end{array}$ when m = n.

Partition of vertices of the type ust of Onm$\begin{array}{} \displaystyle u.s^t \text{of} O.n^m \end{array}$ based on degree product and eccentricity of each vertex when n ≡ 0( mod 2).

New and Modified Eccentric Indices of Octagonal Grid Omn

Published Online: Oct 03, 2018

Page range: 209 - 228

Received: Apr 19, 2018

Accepted: May 26, 2018

DOI: https://doi.org/10.21042/AMNS.2018.1.00016

KeywordsDegree, Eccentricity, Ediz eccentric connectivity index, modified eccentric connectivity index, modified eccentric connectivity polynomial (, ), octagonal grid

© 2018 M. Naeem et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Fig. 1

Partition of vertices of the type ust of Onm$\begin{array}{} \displaystyle u.s^t~~ \text{of}~~ O.n^m \end{array}$ based on degree sum and eccentricity of each vertex when n ≡ 1(mod 2).

Partition of vertices of the type ust of Onm$\begin{array}{} \displaystyle u.s^t~~ \text{of}~~ O.n^m \end{array}$ based on degree product and eccentricity of each vertex when n ≡ 1( mod 2).

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.

Partition of vertices of the type ust of Onm$\begin{array}{} \displaystyle u.s^t~~ \text{of}~~ O.n^m \end{array}$ based on degree sum and eccentricity of each vertex when n ≡ 0( mod 2).

Values of eccentric connectivity index, modified eccentric connectivity index and modified augmented eccentric connectivity index for all possible structures with three and four.

comparison of ξc(Omn), Eξc(Omn) and MAξc(Omn) for Omn,$\begin{array}{} \displaystyle \xi.c(O.m^n), ~ ^E\xi^c(O.m^n)~\text{and}~ ^{MA}\xi^c(O.m^n)~\text{for}~ O.m^n, \end{array}$ when m = n.

Partition of vertices of the type ust of Onm$\begin{array}{} \displaystyle u.s^t~~ \text{of}~~ O.n^m \end{array}$ based on degree product and eccentricity of each vertex when n ≡ 0( mod 2).

New and Modified Eccentric Indices of Octagonal Grid O^m_n

Keywords
Degree, Eccentricity, Ediz eccentric connectivity index, modified eccentric connectivity index, modified eccentric connectivity polynomial (, ), octagonal grid

Partition of vertices of the type ust of Onm$\begin{array}{} \displaystyle u.s^t \text{of} O.n^m \end{array}$ based on degree sum and eccentricity of each vertex when n ≡ 1(mod 2).

Partition of vertices of the type ust of Onm$\begin{array}{} \displaystyle u.s^t \text{of} O.n^m \end{array}$ based on degree product and eccentricity of each vertex when n ≡ 1( mod 2).

Partition of vertices of the type ust of Onm$\begin{array}{} \displaystyle u.s^t \text{of} O.n^m \end{array}$ based on degree sum and eccentricity of each vertex when n ≡ 0( mod 2).

Partition of vertices of the type ust of Onm$\begin{array}{} \displaystyle u.s^t \text{of} O.n^m \end{array}$ based on degree product and eccentricity of each vertex when n ≡ 0( mod 2).