New and Modified Eccentric Indices of Octagonal Grid O^m_n

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)_

Representative	Sust$\begin{array}{} \displaystyle S_{u_s^t} \end{array}$	eccentricity	Range	Frequency
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	4	4n − s + 1	t = 1, n + 1; s = 1	2
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	5	4n − s + 1	t = 1, n + 1,	n − 1
			2 ≤ s ≤n+12$\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	5	3n + s − 1	t = 1, n + 1,	n − 1
			n+12$\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ + 1 ≤ s ≤ n
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	7	4n − 3(s − 1) − t	1 = s,	(n+12$\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ − 1)
			2 ≤ t ≤ n+12$\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	9	4n − 3(s − 1) − t	2 ≤ s ≤ n+12$\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ − 1,	n−34(n+12−1)$\begin{array}{} \displaystyle \frac{ n-3}{4}(\,\,\, \frac{n+1}{2}-1\,\,) \end{array}$
			s + 1 ≤ t ≤ n+12$\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	9	4(n + 1) − s − 3t	2 ≤ s ≤n+12$\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$,	n+14(n+12−1)$\begin{array}{} \displaystyle \frac{ n+1}{4}(\,\,\, \frac{ n+1}{2}-1\,\,) \end{array}$
			2 ≤ t ≤ s
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	9	3n + s − 3t + 2	n+12$\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ + 1 ≤ s ≤ n − 1,	n−14(n−12−1)$\begin{array}{} \displaystyle \frac{ n-1}{4}(\,\,\, \frac{ n-1}{2}-1\,\,) \end{array}$
			2 ≤ t ≤ n + 1 − s
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	9	n + 3s − t − 1	n+12$\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ + 1 ≤ s ≤ n,	n−14(n−12+1)$\begin{array}{} \displaystyle \frac{n-1}{4}(\,\,\, \frac{n-1}{2}+1\,\,) \end{array}$
			n − s + 2 ≤ t ≤n+12$\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	7	3(n − s) + t + 1	s = 1,	(n+12$\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ − 1)
			n+12$\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ + 1 ≤ t ≤ n
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	9	n − s + 3t − 2	2 ≤ s ≤ n+12$\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$,	n−34(n+12−1)$\begin{array}{} \displaystyle \frac{ n-3}{4}(\,\,\, \frac{n+1}{2}-1\,\,) \end{array}$
			n − s + 2 ≤ t ≤ n
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	9	4s − n + t − 3	n+12$\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ + 1 ≤ s ≤ n,	n−14(n+12)$\begin{array}{} \displaystyle \frac{ n-1}{4}(\,\,\, \frac{ n+1}{2}\,\,) \end{array}$
			n+12$\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ + 1 ≤ t ≤ s
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	9	s + 3t − 4	n+12$\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ + 1 ≤ s ≤ n − 1,	n−14(n−12−1)$\begin{array}{} \displaystyle \frac{ n-1}{4}(\,\,\, \frac{ n-1}{2}-1\,\,) \end{array}$
			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	Must$\begin{array}{} \displaystyle M_{u_s^t} \end{array}$	eccentricity	Range	Frequency
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	4	4n − s + 1	t = 1, n + 1; s = 1	2
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	6	4n − s + 1	t = 1, n + 1,	n − 1
			2 ≤ s ≤ n+12$\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	6	3n + s − 1	t = 1, n + 1,	n − 1
			n+12$\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ + 1 ≤ s ≤ n
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	12	4n − 3(s − 1) − t	1 = s,	(n+12$\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ − 1)
			2 ≤ t ≤ n+12$\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	27	4n − 3(s − 1) − t	2 ≤ s ≤ n+12$\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ − 1,	n−34(n+12−1)$\begin{array}{} \displaystyle \frac{ n-3}{4}(\,\,\, \frac{n+1}{2}-1\,\,) \end{array}$
			s + 1 ≤ t ≤ n+12$\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	27	4(n + 1) − s − 3t	2 ≤ s ≤ n+12$\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$,	n+14(n+12−1)$\begin{array}{} \displaystyle \frac{ n+1}{4}(\,\,\, \frac{ n+1}{2}-1\,\,) \end{array}$
			2 ≤ t ≤ s
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	27	3n + s − 3t + 2	n+12$\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ + 1 ≤ s ≤ n − 1,	n−14(n−12−1)$\begin{array}{} \displaystyle \frac{ n-1}{4}(\,\,\, \frac{ n-1}{2}-1\,\,) \end{array}$
			2 ≤ t ≤ n + 1 − s
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	27	n + 3s − t − 1	n+12$\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ + 1 ≤ s ≤ n,	n−14(n−12+1)$\begin{array}{} \displaystyle \frac{n-1}{4}(\,\,\, \frac{n-1}{2}+1\,\,) \end{array}$
			n − s + 2 ≤ t ≤n+12$\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	12	3(n − s) + t + 1	s = 1,	(  n+12$\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ − 1)
			n+12$\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ + 1 ≤ t ≤ n
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	27	n − s + 3t − 2	2 ≤ s ≤ n+12$\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$,	n−34(n+12−1)$\begin{array}{} \displaystyle \frac{ n-3}{4}(\,\,\, \frac{n+1}{2}-1\,\,) \end{array}$
			n − s + 2 ≤ t ≤ n
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	27	4s − n + t − 3	n+12$\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ + 1 ≤ s ≤ n,	n−14(n+12)$\begin{array}{} \displaystyle \frac{ n-1}{4}(\,\,\, \frac{ n+1}{2}\,\,) \end{array}$
			n+12$\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ + 1 ≤ t ≤ s
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	27	s + 3t − 4	n+12$\begin{array}{} \displaystyle \frac{n+1}{2} \end{array}$ + 1 ≤ s ≤ n − 1,	n−14(n−12−1)$\begin{array}{} \displaystyle \frac{ n-1}{4}(\,\,\, \frac{ n-1}{2}-1\,\,) \end{array}$
			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)	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

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	Sust$\begin{array}{} \displaystyle S_{u_s^t} \end{array}$	eccentricity	Range	Frequency
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	4	4n - s + 1	t = 1, n + 1; s = 1	2
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	5	4n − s + 1	t = 1, n + 1,	n
			2 ≤ s ≤ n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 1
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	5	3n + s − 1	t = 1, n + 1,	n − 2
			n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 2 ≤ s ≤ n
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	7	4n − 3(s − 1) − t	s = 1,	n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$
			2 ≤ t ≤ n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 1
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	9	4n − 3(s − 1) − t	2 ≤ s ≤ n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$,	n28−3n4$\begin{array}{} \displaystyle \frac{n^2}{8}-\frac{3n}{4} \end{array}$
			s + 1 ≤ t ≤ n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 1
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	9	4(n + 1) − s − 3t	2 ≤ s ≤ n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$,	n28−n4$\begin{array}{} \displaystyle \frac{n^2}{8}- \frac{ n}{4} \end{array}$
			2 ≤ t ≤ s
n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$	9	3n + s − 3t + 2	n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 1 ≤ s ≤ n − 1,	n28−n4$\begin{array}{} \displaystyle \frac{n^2}{8}- \frac{ n}{4} \end{array}$
			2 ≤ t ≤ n + 1 − s
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	9	n + 3s − t − 1	n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 1 ≤ s ≤ n,	n28−n4$\begin{array}{} \displaystyle \frac{n^2}{8}- \frac{ n}{4} \end{array}$
			n − s + 2 ≤ t ≤ n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 1
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	7	3(n − s) + t + 1	s = 1,	n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ − 1
			n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 2 ≤ t ≤ n
n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$	9	3(n − s) + t + 1	2 ≤ s ≤ n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ − 1,	18$\begin{array}{} \displaystyle \frac{1}{8} \end{array}$(n − 4)(n − 2)
			n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 2 ≤ t ≤ n − s + 1
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	9	n − s + 3t − 2	2 ≤ s ≤ n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$,	n28−n4$\begin{array}{} \displaystyle \frac{n^2}{8}- \frac{ n}{4} \end{array}$
			n − s + 2 ≤ t ≤ n
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	9	4s − n + t − 3	n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 2 ≤ s ≤ n,	n28−n4$\begin{array}{} \displaystyle \frac{n^2}{8}- \frac{ n}{4} \end{array}$
			n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 2 ≤ t ≤ s
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	9	s + 3t − 4	n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 1 ≤ s ≤ n − 1,	n28−n4$\begin{array}{} \displaystyle \frac{n^2}{8}- \frac{ n}{4} \end{array}$
			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	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 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]	ξc(Onm$\begin{array}{} \displaystyle O_n^m \end{array}$)	Eξc(Onm$\begin{array}{} \displaystyle O_n^m \end{array}$)	MAξc(Onm$\begin{array}{} \displaystyle O_n^m \end{array}$)
	[3, 3]	2888	5369165$\begin{array}{} \displaystyle \cfrac{5369}{165} \end{array}$	5880
	[4, 4]	6564	61603915015$\begin{array}{} \displaystyle \cfrac{616039}{15015} \end{array}$	14456
	[5, 5]	13460	746094621322685$\begin{array}{} \displaystyle \cfrac{74609462}{1322685} \end{array}$	32260
	[6, 6]	22972	5635138788580495$\begin{array}{} \displaystyle \cfrac{563513878}{8580495} \end{array}$	56868
	[7, 7]	37280	80347162079310039179150$\begin{array}{} \displaystyle \cfrac{803471620793}{10039179150} \end{array}$	95576
	[8, 8]	55364	136116588596915168440430$\begin{array}{} \displaystyle \cfrac{1361165885969}{15168440430} \end{array}$	144424
	[9, 9]	79784	192924672636118627909300$\begin{array}{} \displaystyle \cfrac{1929246726361}{18627909300} \end{array}$	212136
	[10, 10]	109176	114903717662028710119188365650$\begin{array}{} \displaystyle \cfrac{1149037176620287}{10119188365650} \end{array}$	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	Must$\begin{array}{} \displaystyle M_{u_s^t} \end{array}$	eccentricity	Range	Frequency
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	4	4n − s + 1	t = 1, n + 1; s = 1	2
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	6	4n − s + 1	t = 1, n + 1,	n
			2 ≤ s ≤ n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 1
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	6	3n + s − 1	t = 1, n + 1,	n − 2
			n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 2 ≤ s ≤ n
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	12	4n − 3(s − 1) − t	s = 1,	n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$
			2 ≤ t ≤n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 1
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	27	4n − 3(s − 1) − t	2 ≤ s ≤n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$,	n28−3n4$\begin{array}{} \displaystyle \frac{n^2}{8}-\frac{3n}{4} \end{array}$
			s + 1 ≤ t ≤ ust$\begin{array}{} \displaystyle u_s^t \end{array}$ + 1
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	27	4(n + 1) − s − 3t	2 ≤ s ≤ n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$,	n4(n2−1)$\begin{array}{} \displaystyle \frac{n}{4}(\,\,\, \frac{ n}{2}-1\,\,) \end{array}$
			2 ≤ t ≤ s
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	27	3n + s − 3t + 2	n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 1 ≤ s ≤ n − 1,	n4(n2−1)$\begin{array}{} \displaystyle \frac{n}{4}(\,\,\, \frac{ n}{2}-1\,\,) \end{array}$
			2 ≤ t ≤ n + 1 − s
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	27	n + 3s − t − 1	n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 1 ≤ s ≤ n,	n4(n2+1)$\begin{array}{} \displaystyle \frac{n}{4}(\,\,\, \frac{ n}{2}+1\,\,) \end{array}$
			n − s + 2 ≤ t ≤ n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 1
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	12	3(n − s) + t + 1	s = 1,	n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ − 1
			n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 2 ≤ t ≤ n
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	27	3(n − s) + t + 1	2 ≤ s ≤n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ − 1,	18$\begin{array}{} \displaystyle \frac{1}{8} \end{array}$(n − 4)(n − 2)
			n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 2 ≤ t ≤ n-s + 1
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	27	n − s + 3t − 2	2 ≤ s ≤ n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$,	n4(n2−1)$\begin{array}{} \displaystyle \frac{n}{4}(\,\,\, \frac{ n}{2}-1\,\,) \end{array}$
			n − s + 2 ≤ t ≤ n
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	27	4s − n + t − 3	n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 2 ≤ s ≤ n,	n4(n2−1)$\begin{array}{} \displaystyle \frac{n}{4}(\,\,\, \frac{ n}{2}-1\,\,) \end{array}$
			n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 2 ≤ t ≤ s
ust$\begin{array}{} \displaystyle u_s^t \end{array}$	27	s + 3t − 4	n2$\begin{array}{} \displaystyle \frac{n}{2} \end{array}$ + 1 ≤ s ≤ n − 1,	n4(n2−1)$\begin{array}{} \displaystyle \frac{n}{4}(\,\,\, \frac{ n}{2}-1\,\,) \end{array}$
			s + 1 ≤ t ≤ n