Eccentricity Based Topological Indices of Polygonal Cylinder

S. Sowmya; J. Suthiesh Goldy

1. Introduction

In recent years, graph theory has been used substantially in the branch of mathematical chemistry, owing to its practical applications in solving molecular problems. Over the years, topological indices such as the Wiener index, Balaban index, Hosoya index and Randić index have been studied and significantly improved, and research attention in this area is currently growing exponentially. Throughout this paper, we focus on finite, simple and connected graphs. Let G = (V(G), E(G)) be a graph, where V(G) is the set of all vertices and E(G) is the set of all edges. The degree of a vertex v, denoted by deg(v) or d(v), is the number of edges incident with v in G. The eccentricity ε(v) of a vertex v ∈ V(G) is the maximum distance from v to any other vertex. The goal of this paper is to determine various eccentricity-based indices of the polygonal cylinder. In 2012, Nilanjan De [1] defined the first and second multiplicative Zagreb eccentricity indices as

\prod E_{1} (G) = v \in V (G) \prod ε (v)^{2} and \prod E_{2} (G) = (u, v) \in V (G) \prod ε (u) ε (v) .

(1)

The total eccentricity of a graph G, denoted by ζ(G), is the sum of the eccentricities of all vertices of G. Sharma et al. [2] introduced the eccentric connectivity index, which is defined as

ζ (G) = v \in V (G) \sum d_{u} ε_{u} .

(2)

In 2000, Gupta et al. [3] defined the connective eccentric index as

C^{ξ} (G) = u \in V (G) \sum \frac{d _{u}}{ε _{u}} .

(3)

In 2010, A. R. Ashrafi and M. Ghorbani [4] defined the modified eccentric connectivity index as

ξ_{c} (G) = v \in V (G) \sum (S_{v} ε_{v}),

(4)

where S_v is the sum of the degrees of all vertices adjacent to the vertex v.

The Ediz eccentric connectivity index of G was defined by S. Ediz [5] in 2010 as

^{E} ξ^{c} (G) = v \in V (G) \sum \frac{S _{v}}{ε _{v}} .

(5)

Dureja and Madan [6] introduced the augmented eccentric connectivity index of a graph G, which is defined as

A_{ξ} (G) = v \in V (G) \sum \frac{M _{v}}{ε _{v}} .

(6)

where M_v is the product of the degrees of all neighbours of the vertex v of G.

The modified augmented eccentric connectivity index was proposed by M. Naeem et al. [7] in 2018 and is defined as

^{M A} ξ^{c} (G) = v \in V (G) \sum (M_{v} ε_{v}) .

(7)

The corresponding topological polynomials of Eqs. (4) and (7) are given below. In 2014, N. De et al. [8] defined the modified eccentric connectivity polynomial as

ξ_{c} (G, x) = v \in V (G) \sum S_{u} x^{ε_{u}} .

(8)

The first derivative of Eq. (8) at x = 1 is the modified eccentric connectivity index. In 2018, M. Naeem et al. [7] defined the modified augmented eccentric connectivity polynomial as

^{M A} ξ^{c} (G, x) = v \in V (G) \sum M_{v} x^{ε_{v}} .

(9)

2. Polygonal cylinder

In 2020, Abdul Rauf Nizami et al. [9] defined the polygonal cylinder C_n,m. Consider two copies of the path P_n, n ≥ 3, with vertices u₁, u₂, …, u_n and v₁, v₂, …, v_n, respectively. In the Cartesian product P_n × P_n, identify the vertices (u₁, v₁), (u₁, v₂), …, (u₁, v_n) with the vertices (u_n, v₁), (u_n, v₂), …, (u_n, v_n), and the edge joining (u₁, v_i) and (u₁, v_i+1) with the edge joining (u_n, v_i) and (u_n, v_i+1), where 1 ≤ i ≤ n. The resulting graph is the polygonal cylinder, or (n − 1)-gonal cylinder, denoted by C_n,n. The graph C_n,n has n(n − 1) vertices and (2n − 1)(n − 1) edges. Figure 1 shows the grid P₄ × P₄, and Figure 2 shows the 3-gonal cylinder C_4,4.

The path P4 with vertices u1 to u4 drawn above the 4 by 4 grid P4 × P4, whose sixteen vertices are labelled 11 to 44. — Fig. 1. Grid P₄ × P₄

The vertical path P4 with vertices v1 to v4 beside the polygonal cylinder C4,4, drawn as three columns of vertices labelled 11=41 to 11=44, 21 to 24 and 31 to 34, joined by straight and curved edges. — Fig. 2. Polygonal cylinder C₄,₄

3. Eccentricity-based indices of the polygonal cylinder

In this section, we determine various eccentricity-based indices of the polygonal cylinder and their respective polynomials.

3.1. First multiplicative Zagreb eccentricity index of the polygonal cylinder Cₙ,ₙ

If n is even, C_n,n has n(n − 1) vertices, which form (n/2) sets of (2n − 2) vertices with eccentricity (n − 1 + k), for k = 0 to (n/2 − 1). Using Eq. (1), we have

\prod E_{1} (G) = v \in V (G) \prod ε (v)^{2} = ε (u_{1})^{2} \cdot ε (u_{2})^{2} \dots ε (u_{n - 1})^{2} \cdot ε (v_{1})^{2} \dots ε (v_{n})^{2} = (n - 1)^{2} (n - 1 + 1)^{2} (n - 1 + 2)^{2} \dots (n - 1 + (\frac{n}{2} - 1))^{2} = k = 0 \prod \frac{n}{2} - 1 ((n - 1 + k)^{2})^{2 n - 2} .

(10)

This result is true for all n ≥ 4. If n is odd, (n − 1) vertices have eccentricity (n − 1), and ⌊n/2⌋ sets of (2n − 2) vertices have eccentricity (n + k), for k = 0 to (⌊n/2⌋ − 1). Using Eq. (1), we have

\prod E_{1} (G) = v \in V (G) \prod ε (v)^{2} = ((n - 1)^{2})^{n - 1} (n)^{2} (n + 1)^{2} (n + 2)^{2} \dots (n + (⌊ \frac{n}{2} ⌋ - 1))^{2} = ((n - 1)^{2})^{n - 1} k = 0 \prod ⌊ \frac{n}{2} ⌋ - 1 ((n + k)^{2})^{2 n - 2} .

(11)

This result is true for all n ≥ 5. Hence,

\prod E_{1} (G) = ⎩ ⎨ ⎧ k = 0 \prod \frac{n}{2} - 1 ((n - 1 + k)^{2})^{2 n - 2}; ((n - 1)^{2})^{n - 1} k = 0 \prod ⌊ \frac{n}{2} ⌋ - 1 ((n + k)^{2})^{2 n - 2}; if n is even if n is odd .

(12)

3.2. Second multiplicative Zagreb eccentricity index of the polygonal cylinder Cₙ,ₙ

For even n, the second multiplicative Zagreb eccentricity index can be computed as follows: ⌊(n − 1)/2⌋ sets of (2n − 2) vertices have eccentricity (n + k − 1), for k = 1 to ⌊(n − 1)/2⌋; (3n − 3) vertices have eccentricity (n − 1); and ⌊(n − 1)/2⌋ sets of (2n − 2) vertices have eccentricity of the form ε(u)ε(v) = (n + k − 1)(n + k − 2), for k = 1 to ⌊(n − 1)/2⌋. Using Eq. (1), we have

\prod E_{2} (G) = (u, v) \in V (G) \prod ε (u) ε (v) = k = 1 \prod ⌊ \frac{n - 1}{2} ⌋ ((n + k - 1) (n + k - 1))^{2 n - 2} ((n - 1) (n - 1))^{3 n - 3} k = 1 \prod ⌊ \frac{n - 1}{2} ⌋ ((n + k - 1) (n + k - 2))^{2 n - 2} = k = 1 \prod ⌊ \frac{n - 1}{2} ⌋ ((n + k - 1)^{2})^{2 n - 2} ((n - 1)^{2})^{3 n - 3} k = 1 \prod ⌊ \frac{n - 1}{2} ⌋ ((n + k - 1) (n + k - 2))^{2 n - 2}

(13)

This is true for all n ≥ 4. Similarly, the result follows for odd n ≥ 5. Hence,

\prod E_{2} (G) = ⎩ ⎨ ⎧ k = 1 \prod ⌊ \frac{n - 1}{2} ⌋ ((n - 1 + k)^{2})^{2 n - 2} ((n - 1)^{2})^{3 n - 3} k = 1 \prod ⌊ \frac{n - 1}{2} ⌋ ((n + k - 1) (n + k - 2))^{2 n - 2}; k = 1 \prod ⌊ \frac{n - 1}{2} ⌋ ((n - 1 + k)^{2})^{2 n - 2} ((n - 1)^{2})^{n - 1} k = 1 \prod ⌊ \frac{n - 1}{2} ⌋ ((n + k - 1) (n + k - 2))^{2 n - 2}; if n is even if n is odd .

(14)

3.3. Total eccentricity index of the polygonal cylinder Cₙ,ₙ

For even n, C_n,n has (n/2) sets of (2n − 2) vertices with eccentricity (n + 1 − k), k = 0 to (n/2 − 1). Similarly, for odd n, it has ⌊n/2⌋ sets of (2n − 2) vertices with eccentricity (n + k), k = 0 to (⌊n/2⌋ − 1), and (n − 1) vertices with eccentricity (n − 1). By the definition of total eccentricity, we have

ζ (G) = v \in V (G) \sum ε_{u} = ⎩ ⎨ ⎧ (2 n - 2) k = 0 \sum \frac{n}{2} - 1 (n + k - 1); (n - 1)^{2} + (2 n - 2) k = 0 \sum ⌊ \frac{n}{2} ⌋ - 1 (n + k); if n is even if n is odd .

(15)

Hence the result.

3.4. Eccentric connectivity index and connective eccentric index of the polygonal cylinder Cₙ,ₙ

Using Eq. (2), we have ξ(G) = ∑_v∈V(G) d_uε_u. For even n, (2n − 2) vertices of this graph have degree 3 and eccentricity (n + n/2 − 2), and (n/2 − 1) sets of (2n − 2) vertices have degree 4 and eccentricity (n + k − 1), k = 0 to (n/2 − 2). Similarly, for odd n, (2n − 2) vertices have degree 3 and eccentricity (n + ⌊n/2⌋ − 1); (⌊n/2⌋ − 1) sets of (2n − 2) vertices have degree 4 and eccentricity (n + k), k = 0 to (⌊n/2⌋ − 2); and 4 vertices have degree (n − 1) and eccentricity (n − 1). Hence,

ξ (G) = ⎩ ⎨ ⎧ (2 n - 2) [3 \cdot (n + \frac{n}{2} - 2)] + (2 n - 2) k = 0 \sum \frac{n}{2} - 2 [4 \cdot (n + k - 1)]; (2 n - 2) [3 \cdot (n + ⌊ \frac{n}{2} ⌋ - 1)] + 4 \cdot (n - 1) (n - 1) + (2 n - 2) k = 0 \sum ⌊ \frac{n}{2} ⌋ - 2 [4 \cdot (n + k)]; if n is even if n is odd .

(16)

Using the above index, we can compute the connective eccentric index. By Eq. (3), we have

C^{ξ} (G) = v \in V (G) \sum \frac{d _{u}}{ε _{u}} = ⎩ ⎨ ⎧ (2 n - 2) \frac{3}{( n + \frac{n}{2} - 2 )} + (2 n - 2) k = 0 \sum \frac{n}{2} - 2 [\frac{4}{( n + k - 1 )}]; (2 n - 2) \frac{3}{( n + ⌊ \frac{n}{2} ⌋ - 1 )} + 4 [\frac{( n - 1 )}{( n - 1 )}] + (2 n - 2) k = 0 \sum ⌊ \frac{n}{2} ⌋ - 2 [\frac{4}{( n + k )}]; if n is even if n is odd .

(17)

Hence the result.

3.5. Modified eccentric connectivity index and Ediz eccentric connectivity index of the polygonal cylinder Cₙ,ₙ

From the definition of ξ_c(G), S_v is the sum of the degrees of all vertices adjacent to the vertex v. For odd n, (2n − 2) vertices have S_v = 3 + 3 + 4 = 10 and eccentricity (n + ((n − 5)/2) + 1); (2n − 2) vertices have S_v = 3 + 4 + 4 + 4 = 15 and eccentricity ((n − 5)/2); and (n − 1) vertices have S_v = 4 + 4 + 4 + 4 = 16 and eccentricity (n − 1). Similarly, for even n, (2n − 2) vertices have S_v = 3 + 3 + 4 = 10 and eccentricity (n + ⌊(n − 3)/2⌋); (2n − 2) vertices have S_v = 3 + 4 + 4 + 4 = 15 and eccentricity (n + ⌊(n − 3)/2⌋ − 1); and (n − 5) sets of (2n − 2) vertices have S_v = 4 + 4 + 4 + 4 = 16 and eccentricity ((3n − 8)/2 − k), k = 0 to (n − 6). Hence the result. The corresponding topological polynomial of Eq. (4) is Eq. (8), ξ_c(G, x) = ∑_v∈V(G) S_vx^ε_v. Using the above data, we get

ξ^{c} (G, x) = ⎩ ⎨ ⎧ (2 n - 2) (10 x^{(n + (\frac{n - 5}{2}) + 1)}) + (2 n - 2) (15 x^{(n + (\frac{n - 5}{2}))}) + (n - 1) (16 x^{(n - 1)}); (2 n - 2) (10 x^{(n + ⌊ \frac{n - 3}{2} ⌋)}) + (2 n - 2) (15 x^{(n + ⌊ \frac{n - 3}{2} ⌋ - 1)}) + (2 n - 2) k = 0 \sum n - 6 (16 x^{(\frac{3 n - 8}{2} - k)}); if n is odd if n is even .

(18)

We can compute the Ediz eccentric connectivity index ^Eξ^c(G) by using the modified eccentric connectivity index of C_n,n. By Eq. (5), we have

^{E} ξ^{c} (G) = v \in V (G) \sum \frac{S _{v}}{ε _{v}} = ⎩ ⎨ ⎧ (2 n - 2) (\frac{10}{( n + ( \frac{n - 5}{2} ) + 1 )}) + (2 n - 2) (\frac{15}{( n + ( \frac{n - 5}{2} ) )}) + (n - 1) (\frac{16}{( n - 1 )}); (2 n - 2) (\frac{10}{( n + ⌊ \frac{n - 3}{2} ⌋ )}) + (2 n - 2) (\frac{15}{( n + ⌊ \frac{n - 3}{2} ⌋ - 1 )}) + (2 n - 2) k = 0 \sum n - 6 (\frac{16}{\frac{3 n - 8}{2} - k}); if n is odd if n is even .

(19)

Hence the result.

3.6. Augmented eccentric connectivity index, modified augmented eccentric connectivity index and its polynomial of the polygonal cylinder Cₙ,ₙ

By Eq. (6), we have A_ξ(G) = ∑_v∈V(G) M_v/ε_v. For odd n, (2n − 2) vertices have M_v = 3·3·4 = 36 and eccentricity (n + ((n − 5)/2) + 1); (2n − 2) vertices have M_v = 3·4·4·4 = 192 and eccentricity (n + ((n − 5)/2)); ((n − 5)/2) sets of (2n − 2) vertices have M_v = 4·4·4·4 = 256 and eccentricity (n + k), k = 0 to ((n − 7)/2); and (n − 1) vertices have M_v = 4·4·4·4 = 256 and eccentricity (n − 1). Similarly, for even n, (2n − 2) vertices have M_v = 3·3·4 = 36 and eccentricity (n + ⌊(n − 3)/2⌋); (2n − 2) vertices have M_v = 3·4·4·4 = 192 and eccentricity (n + ⌊(n − 3)/2⌋ − 1); and (n − 5) sets of (2n − 2) vertices have M_v = 4·4·4·4 = 256 and eccentricity ((3n − 8)/2 − k), k = 0 to (n − 6). Hence,

A_{ξ} (G) = ⎩ ⎨ ⎧ (2 n - 2) (\frac{36}{n + ( \frac{n - 5}{2} ) + 1}) + (2 n - 2) (\frac{192}{n + ( \frac{n - 5}{2} )}) + (2 n - 2) k = 0 \sum \frac{n - 7}{2} (\frac{256}{n + k}) + (n - 1) (\frac{256}{n - 1}); (2 n - 2) (\frac{36}{n + ⌊ \frac{n - 3}{2} ⌋}) + (2 n - 2) (\frac{192}{n + ⌊ \frac{n - 3}{2} ⌋ - 1}) + (2 n - 2) k = 0 \sum n - 6 (\frac{256}{( \frac{3 n - 8}{2} - k )}); if n is odd if n is even

(20)

Using the above result, we can compute the modified augmented eccentric connectivity index. By Eq. (7), we have

^{M A} ξ^{c} (G) = v \in V (G) \sum M_{v} ε_{v} = ⎩ ⎨ ⎧ (2 n - 2) (36 (n + (\frac{n - 5}{2}) + 1)) + (2 n - 2) (192 (n + (\frac{n - 5}{2}))) + (2 n - 2) k = 0 \sum \frac{n - 7}{2} (256 (n + k)) + (n - 1) (256 (n - 1)); (2 n - 2) (36 (n + ⌊ \frac{n - 3}{2} ⌋)) + (2 n - 2) (192 (n + ⌊ \frac{n - 3}{2} ⌋ - 1)) + (2 n - 2) k = 0 \sum n - 6 (256 (\frac{3 n - 8}{2} - k)); if n is odd if n is even

(21)

Also, we can compute the modified augmented eccentric connectivity polynomial ^MAξ^c(G, x) using its index ^MAξ^c(G). Therefore,

^{M A} ξ^{c} (G, x) = v \in V (G) \sum M_{v} x^{ε_{v}} = ⎩ ⎨ ⎧ (2 n - 2) (36 x^{(n + (\frac{n - 5}{2}) + 1)}) + (2 n - 2) (192 x^{(n + (\frac{n - 5}{2}))}) + (2 n - 2) k = 0 \sum \frac{n - 7}{2} (256 x^{(n + k)}) + (n - 1) (256 x^{(n - 1)}); (2 n - 2) (36 x^{(n + ⌊ \frac{n - 3}{2} ⌋)}) + (2 n - 2) (192 x^{(n + ⌊ \frac{n - 3}{2} ⌋ - 1)}) + (2 n - 2) k = 0 \sum n - 6 (256 x^{(\frac{3 n - 8}{2} - k)}); if n is odd if n is even

(22)

Hence the result.

Acknowledgements

We thank the referees for their careful reading and helpful suggestions, which led to many improvements.

References

De N. On multiplicative Zagreb eccentricity indices. South Asian Journal of Mathematics. 2012;2(6):570-7.
Sharma V, Goswami R, Madan AK. Eccentric connectivity index: a novel highly discriminating topological descriptor for structure-property and structure-activity studies. Journal of Chemical Information and Computer Sciences. 1997;37(2):273-82. https://doi.org/10.1021/ci960049h
Gupta S, Singh M, Madan AK. Connective eccentricity index: a novel topological descriptor for predicting biological activity. Journal of Molecular Graphics and Modelling. 2000;18(1):18-25. https://doi.org/10.1016/S1093-3263(00)00027-9
Ashrafi AR, Ghorbani M. A study of fullerenes by MEC polynomials. Electronic Materials Letters. 2010;6(2):87-90. https://doi.org/10.3365/eml.2010.06.087
Ediz S. Computing Ediz eccentric connectivity index of an infinite class of nanostar dendrimers. Optoelectronics and Advanced Materials – Rapid Communications. 2010;4:1847-8.
Dureja H, Madan AK. Superaugmented eccentric connectivity indices: new-generation highly discriminating topological descriptors for QSAR/QSPR modeling. Medicinal Chemistry Research. 2007;16(7-9):331-41. https://doi.org/10.1007/s00044-007-9032-9
Naeem M, Siddiqui MK, Guirao JLG, Gao W. New and modified eccentric indices of octagonal grid Oₙᵐ. Applied Mathematics and Nonlinear Sciences. 2018;3(1):209-28. https://doi.org/10.21042/AMNS.2018.1.00016
De N, Pal A, Nayeem SMA. Modified eccentric connectivity of generalized thorn graphs. International Journal of Computational Mathematics. 2014;2014:436140. https://doi.org/10.1155/2014/436140
Nizami AR, Naseem A, Ahmed HMW. The polygonal cylinder and its Hosoya polynomial. Online Journal of Analytic Combinatorics. 2020;(15):1-12. https://doi.org/10.61091/ojac-1509

Eccentricity Based Topological Indices of Polygonal Cylinder

Abstract

Keywords

1. Introduction

2. Polygonal cylinder

3. Eccentricity-based indices of the polygonal cylinder

3.1. First multiplicative Zagreb eccentricity index of the polygonal cylinder Cₙ,ₙ

3.2. Second multiplicative Zagreb eccentricity index of the polygonal cylinder Cₙ,ₙ

3.3. Total eccentricity index of the polygonal cylinder Cₙ,ₙ

3.4. Eccentric connectivity index and connective eccentric index of the polygonal cylinder Cₙ,ₙ

3.5. Modified eccentric connectivity index and Ediz eccentric connectivity index of the polygonal cylinder Cₙ,ₙ

3.6. Augmented eccentric connectivity index, modified augmented eccentric connectivity index and its polynomial of the polygonal cylinder Cₙ,ₙ

Acknowledgements

References

Related articles

Tensor Sum Operator and Dynamical System in Measurable Function Spaces

The Hop Domination Number of Comb Product Graphs

Vertex Polynomial for Switching in Various Graphs

Open Access & Licensing

Related articles

Tensor Sum Operator and Dynamical System in Measurable Function Spaces
G. S. Vigneswari, P. Chandra Kala, P. Sujin
Let X be a Hausdorff topological space and let B(E) be the Banach algebra of all bounded linear operators on a Banach space E. Let…
10 Jun 2026

The Hop Domination Number of Comb Product Graphs
D. Anusha, S. Joseph Robin
A set S ⊆ V of a graph G is a hop dominating set of G if for every v ∈ V − S, there exists u ∈ S such that d(u, v) = 2. The minimu…
03 Jun 2026

Vertex Polynomial for Switching in Various Graphs
P. Maya, G. R. Sanma
The vertex polynomial of a graph G is defined as V(G, x) = Σ vₖxᵏ (k = 0 to Δ(G)), where Δ(G) = max{d(v) / v ∈ V} and vₖ is the nu…
11 Feb 2026