Computing Zagreb Connection Indices for the Cartesian Product of Path and Cycle Graphs

Aiman Arshad1*, and Aneeta Afzal2

1Department of Mathematics, School of Science, University of Management and Technology, Lahore, Pakistan

2Department of Physics, University of Gujrat, Pakistan

Original Article Open Access
DOI: https://doi.org/10.32350/sir.83.05

Abstract

Topological indices (TIs) are a class of graph-based descriptors widely used in chemiformatics and Quantitative Structure-Activity Relationship (QSAR) studies. TIs capture key structural features of molecules by encoding graph-theoretic information, offering a quantitative representation of molecular topology. Each mathematical network is associated with a specific numerical value determined by a TI function. Among TIs, the Zagreb connection indices (ZCIs) have been extensively researched. This study delves into the Cartesian product of cycle graphs with path graphs, elucidating the comprehensive implications of ZCIs. These indices encompass the first ZCI(FZCI), second ZCI(SZCI), and third ZCI(TZCI). Furthermore, comprehensive results of the modified first ZCI(MFZCI), modified second ZCI(MSZCI), and modified third ZCI(MTZCI) are presented, along with the first multiplicative ZCI (FMZCI), second multiplicative ZCI (SMZCI), and third multiplicative ZCI (TMZCI). Moreover, modified first multiplicative ZCI (MFMZCI), modified second multiplicative ZCI (MSMZCI), and modified third multiplicative ZCI (MTMZCI) are also calculated. To provide precision, both the graphical and numerical analyses of the computed findings are aligned for the two Cartesian products.

Keywords: Cartesian product, cycle graph, path graph, topological index (TI), Zagreb index (ZI), Zagreb connection index (ZCI)
*Corresponding author: [email protected]

Published: 26-09-2024

1. Introduction

A numerical representation is the mathematical encoding of a molecular graph that predicts the biological, structural, physicochemical, and toxicological properties of a chemical compound. These representations are also useful to investigate correlation values among various octane isomers [1]. Topological indices (TIs) are utilized to study the medical behavior of drugs, crystalline materials, and nanomaterials which hold significant importance in chemical and pharmaceutical industries [2]. [3–5] noted that transition indices find extensive application in researching Quantitative Structure-Activity Relationships (QSARs) and Quantitative Structure-Property Relationships (QSPRs). These relationships are crucial within the field of cheminformatics [4, 5]. The graph can be categorized into three types, namely distance, polynomial, and degree, each representing a different classification of . When classification is based on distance, it is referred to as distance-based . Wiener initially presented distance-based [6] in his 1947 study on the boiling point of paraffin, which
is now commonly referred to as the Wiener index. The utilization of and graph invariants, derived from the distances between vertices in graphs, is prevalent in the field of mathematical chemistry [7].

Among the various classes of TIs, the degree-based class holds particular prominence, relying on vertex degrees. This category is further divided into two sub-classes, namely degree-based TIs and connection-based TIs. Gutman and Trinajstic [1] introduced these TIs to calculate the total -electron energy of alternant hydrocarbons. These indices utilize the first Zagreb index, a well-established topological index. Additionally, Gutman and Furtula later defined two other indexes, namely the second Zagreb index (ZCIs) and the third Zagreb index (TZI) [8, 9]. In 2003, [10] examined a novel index known as the modified. Hao [11] conducted a thorough comparison of these introduced , carefully evaluating the outcomes associated with these indices. Additionally, Das [12] delved into the investigation of various modified Zagreb indices (ZCIs) related to graph operations.

Recently, Ali and Trinnajstic [13] introduced a novel approach to investigate the psychochemical properties of compounds. This method involves the introduction of the connection number CN of the vertex and the initiation of Zagreb connection indices . The number of those vertices which are at distance two from a certain vertex is said to be a CN of that vertex. They reported that the newly proposed connection-based (ZCIs) have a better ability to forecast the psychochemical properties of various molecular structures instead of classical .
After the introduction of , numerous researchers embarked on investigating novel connection-based indices. Du et al. [14] applied a connection-based modified first Zagreb index ( ) to identify extremal alkanes. In 2021, Sattar and Javaid [15] derived general expressions for calculating the modified Zagreb connection index (MZCIs ) of dendrimer nanostars. Additionally, in 2020, Ali and Javaid undertook the task of computing MZCIs for T-sum graphs. Recently, Arshad et al. [16] computed the result of the Cartesian product of path and complete graphs.

Rene Descartes is credited with inventing Cartesian coordinates in the 17th century, which revolutionized mathematics by establishing the first systematic connection between geometry and algebra. His analytic geometry, which gave rise to the notion, is honored by the moniker Cartesian product. The two branches of mathematics that gave rise to the concept of the Cartesian product are set theory created by Georg Cantor and analytic geometry invented by Rene Descartes. The Cartesian product of two graphs was first developed by Whitehead and Russell in 1912. Later, they were periodically /developed, most notably in 1959 by Sabidussi [17]. In 2000, Imrich et al. [18] investigated various distinct forms of network's Cartesian products. With this progression, Imrich and Peterin [19] continued to figure the TIs of the Cartesian products of graphs. Following the introduction of ZCIs, several researchers have calculated the Cartesian product of two networks attributes using . Vizing [20] discovered the Cartesian product of two distinct graphs in 1963. In 2017, Shakila and Imran also investigated degree-based TIs for the Cartesian product of connected graphs with F-sums.

This paper presents precise solutions for connection-based fields (ZCIs) in the Cartesian product of path and cycle graphs. The manuscript is structured as follows. Section 2 covers elementary definitions to assist readers comprehend the core concepts. Section 3 contains the general expressions to calculate the ZCIs of the Cartesian product between cycle graphs and path graphs. Finally, Section 4 presents the key findings and Section 5 states the conclusions, respectively.

2. PRELIMINARIES

This section includes key primary definitions from the literature that are essential to comprehend the main findings presented in this manuscript.

Definition 2.1.[21]Let A = E(A), V(A) be a graph, where E(A) and V(A) represent the collection of edges and vertices, respectively. Following that, the degree-based ZIs are defined as

1. Z1 (A) = ∑(h∈V(A)  (dA (h)2 = ∑(h∈V(A) (dA (h)+dA (q),

2. Z2 (A) = ∑(h∈V(A) (dA (h)+dA (q),

where dA (h) and dA (q) show the degree of the vertices h and q, respectively.

Definition 2.2. [13] For a graph A,FZCI and SZCI are given as

1. Z1 CI(A)=∑(h∈V(A) (τA (h)2,

2. Z2 CI(A)=∑(h∈V(A) (τA (h)×τA (q),

where τA (h) and τA (q) represent the CN of the vertices h and q, respectively.

Definition 2.3.[22] For a graph A the MFZCIs, MSZCIs, and MZCIs can be given as

1. Z1 C* I(A)=∑(h∈V(A) (τA (h)+τA (q),

2. Z2 C* I(A)=∑(h∈V(A) [dA (h)τA (q)+dA (q)τA (h)],

3. Z3 C* I(A)=∑(h∈V(A) [dA (h)τA (h)+dA (q)τA (q)].

Definition 2.4.[23] For a graph A, FMZCIs, SMZCIs, TMZCIs, and FMZCIs can be as

MZ1 CI(A)=∏(h∈V(A) τA (h)2,

MZ2 CI(A)=∏(h∈V(A) (τA (h)×τA (q),

MZ3 CI(A)=∏(h∈V(A) (dA (h)τA (h),

MZ4 CI(A)=∏(h∈V(A) (τA (h)+τA (q).

Definition 2.5. [22] For a graph A, MFMZCIs, MSMZCIs, and MFMZCIs can be defined as

MZ1 C* I(A) = ∏(h∈V(A) [dA (h)τA (q) +dA (q)τA (h)],

MZ2 C* I(A)=∏(h∈V(A) [dA (h)τA (h) +dA (q)τA (q)],

MZ3 C* I(A)= ∏(h∈V(A) [ dA (h)τA (h) ×dA (q)τA (q)].

3. COMBINING CYCLE GRAPHS WITH PATH GRAPHS USING CARTESIAN PRODUCT

Let A1≅Cm be a cycle network of order m, denoted by V(A1 )= vs:1≤s≤m and

E(A1 )=vs ts:1≤s≤m and 1≤t≤m but s≠t.

Now, let A2≅Pn be a path network of order n, V(A2 )=xs:1≤s≤n and E(A2 )=xs x_(s+1):1≤s≤n-1.

The Cartesian product of A1×A2 ≅ Cm ×Pn. The edge set, denoted by E(A1 ) × E(A2 ) is defined as follows:

(vs,xA ) is adjacent (vt,xb ) if

vs=vt and xA∼xb,

xA=xb and vs∼vt.

This section explores the Cartesian product of the cycle graph Cm with the path graph Pn, where ( m≥4 and n≥6 ) are shown in Figure 1. Moving on to Figure 2, we delve into the structure of the Cartesian product involving the cycle graph C4 and the path graph P6.


Figure 1. C4 and P6

At the initial level, the first layer of C4 connects to the last edges of P6, resulting in a CN of 3. The second layer of C4 connects to the last layer of P6, resulting in a CN of 4. Furthermore, the process is repeated and the 3rd, 4th, and 5th layers result in a CN of 4, although the last (6th) layer of CN is the same as the first layer. We have also labeled each edge and vertex with their degree and these CN are clearly indicated in Figure 2.

Tables 1 and 2 separately present the edge partition based on degree and . Tables 3 and 4 present the vertex partition based on degree and connection.


Figure 2. Cartesian Product of (C4 and P6 )

Now, we partition the vertices and edges based on the connection and degree number of the Cartesian product with cycle graphs and path graphs.

Table 1. Degree-based Edge Partition

S.R

 E_(d d(n1 ),d(n2 )

Number of edges

1

 E3,3d 2m

2

 E3,4d 2m

3

 E4,4d 2mn-5m

Table 2. Connection-based Edge Partition

S.R

 E_(c τ(n1 ),τ(n2)

Number of edges

1

 E4,4c 2m

2

 E4,6c 2m

3

 E6,6c 2m

4

 E6,7c 2m

5

 E7,7c 2mn-9m

Table 3. Degree-based Vertex Partition

S.R

 V_(d d(n1 ),d(n2 )

Number of vertices

1

 V3d 2m

2

 V4d m(n-2)

Table 4. Connection-based Vertex Partition of

S.R

 V_(c τ(n1 ),τ(n2 )

Number of vertices

1

 V4c 2m

2

 V6c 2m

3

 V7c m(n-4)

4. RESULTS

Let A ≅ Cm × Pn be a graph obtained by a Cartesian product of the cycle graph (Cm) and path graph (Pn), where m ≥ 4 and n ≥ 6. Now, in this section we compute the main results for a graph A.

Theorem 4.1. For a graph A, the FZCI is given by

FZCI (A)=49mn -92m.

Proof: Using Definition 2.2, we have

Z1 CI(A) = ∑(h∈V(A)  [τA (h)]2

= ∑_(4c)  [τA (h)]2 +∑_(h∈V6c)  [τA (h)]2 + ∑_(h∈V7c)  [τA (h)]2

=|V4c |(4)2 +|V6c | (6)2+|V7c |(7)2

=(2m) (4)2+ (2m) + m (n-4) (7)2

=49mn -92m.

Theorem 4.2. For a graph A, the SZCI is given by

SZCI (A)= 98mn -205m.

Proof: Using Definition 2.2, we have

Z2 CI(A) = ∑(h∈V(A)  [τA (h)×τA (q)]

= ∑_(hq∈E4,4c)  [τA (h)×τA (q)] +∑_(hq∈E4,6c)  [τA (h)×τA (q)] +∑_(hq∈E6,6c)  [τA (h)×τA (q)]+∑_(hq∈E6,7c)  [τA (h)×τA (q)] +∑_(hq∈E7,7c)  [τA (h)×τA (q)]

=|E4,4c |(4×4) +|E4,6c |(4×6) +|E6,6c |(6×6) +|E6,7c |(6×7) +|E7,7c |(7×7)

=2m(16) +2m(24) +2m(36) +2m(42) + (2mn-9m) (49)

= 98mn - 205m.

Theorem 4.3. For a graph A, the MFZCI is given by

MFZCI (A)=28mn-40m.

Proof. Using Definition 2.3

Z1 CI* (A) = ∑(h∈V(A)  [τA (h)+τA (q)]

=∑_(hq∈E4,4c)  [τA (h)+τA (q)] + ∑_(hq∈E4,6c)  [τA (h)+τA (q)] + ∑_(hq∈E6,6c)  [τA (h)+τA (q)]+∑_(hq∈E6,7c)  [τA (h)+τA (q)] + ∑_(hq∈E7,7c)  [τA (h)+τA (q)].

= |E4,4c | (4+4) + |E4,6c | (4+6) + +|E6,6c | +|E6,6c |(6+6) + |E6,7c | (6+7) + |E7,7c | (7+7)

=2m(8) + 2m(10) + 2m(13) + (2mn-9m)(14)

= 28mn - 40m.

Theorem 4.4. For a graph A, the MSZCI is given by

MSZCI (A)= 112mn -188m.

Proof: Using Definition 2.3, we have

Z2 C* I(A) = ∑(h∈V(A) [dA (h)τA (q) + dA (q)τA (h)]

= ∑(hq∈(E4,4c)/(E3,3d ) [ dA (h)τA (q) + dA (q)τA (h) ] + ∑_(hq∈(E4,6c)/(E3,4d ) [ dA (h)τA (q) + dA (q)τA (h) ] + ∑(hq∈(E6,6c)/(E4,4d ) [ dA (h)τA (q) + dA (q)τA (h) ] + ∑(hq∈(E6,7c)/(E4,4d ) [ dA (h)τA (q) + dA (q)τA (h) ]+ ∑_(hq∈(E7,7c)/(E4,4d ) [ dA (h)τA (q) + dA (q)τA (h) ]

= |(E4,4c)/(E3,3d )| [ (4×3) + (4×3) ] + |(E4,6c)/(E3,4d )| [ (3×6) + (4×4) ] + |(E6,6c)/(E4,4d )| [ (6×4) + (6×4) ] + |(E6,7c)/(E4,4d )| [ (7×4) + (4×6) ] + |(E7,7c)/(E4,4d )| [ (7×4) + (7×4) ]

= 2m(24) + 2m(34) + 2m(48) + 2m(52) + (2mn-9m)(14)

= 112mn - 188m.

Theorem 4.5. For a graph A, the MTZCI is given by

MTZCI (A) = 112mn - 184m.

Proof: Using Definition 2.3, we have

Z3 C* I(A) = ∑(h∈V(A) [dA (h)τA (h)+ dA (q)τA (q) ]

= ∑(hq∈(E4,4c)/(E3,3d ) [ dA (h)τA (h) + dA (q)τA (q) ] + ∑_(hq∈(E4,6c)/(E3,4d ) [ dA (h)τA (h) + dA (q)τA (q) ] + ∑(hq∈(E6,6c)/(E4,4d ) [ dA (h)τA (h) + dA (q)τA (q) ] + ∑(hq∈(E6,7c)/(E4,4d ) [ dA (h)τA (h) + dA (q)τA (q) ] + ∑_(hq∈(E7,7c)/(E4,4d ) [ dA (h)τA (h) + dA (q)τA (q) ]

= |(E4,4c)/(E3,3d )| [ (4×3) + (4×3) ] + |(E4,6c)/(E3,4d )| [ (4×3) + (6×4) ] + |(E6,6c)/(E4,4d )| [ (6×4) + (6×4) ] + |(E6,7c)/(E4,4d )| [ (6×4) + (7×4) ] + |(E7,7c)/(E4,4d )| [ (7×4) + (7×4) ]

= 2m(24) + 2m(36) + 2m(48) + 2m(52) + (2mn-9m)(56)

= 112mn - 184m.

Theorem 4.6. For a graph A, the FMZCI is given by

FMZCI (A) = 112896m3 n - 451584m3.

Proof: Using Definition 2.4, we have

MZ1 CI(A) = ∏(h∈V(A)  [τA (h)]2

= ∏_(4c)  [τA (h)]2 × ∏(4c) [τA (h)]2 × ∏_(4c)  [τA (h)]2 × ∏4c)  [τA (h)]2 × ∏_(4c)  [τA (h)]2

= |V4c |(4)2 × |V6c |(6)2 × |V7c |(7)2

=(2m)(4)2 × (2m)(6)2 × m(n-4)(7)2

=112896m3 n - 451584m3.

Theorem 4.7. For a graph A, the SMZCI is given by

SMZCI (A)= 9289728m5 n - 4096770084m.5

Proof: Using Definition 2.4, we have

MZ2 CI(A) = ∏(h∈V(A)  [τA (h) × τA (q)]

= ∏(hq∈E4,4c)  [τA (h) × τA (q)] × ∏(hq∈E4,6c)  [τA (h) × τA (q)] × ∏(hq∈E6,6c)  [τA (h) × τA (q)] × ∏_(hq∈E6,7c)  [τA (h) × τA (q)] × ∏_(hq∈E7,7c)  [τA (h) × τA (q)].

= |E4,4c | (4×4) × |E4,6c |(4×6) × |E6,6c |(6×6) × |E6,7c |(6×7) × |E7,7c |(7×7)

= 2m(16) + 2m(24) + 2m(36) + 2m(42) + (2mn-9m)(49)

= 9289728m5 n -4096770084m5.

Theorem 4.8. For a graph A, the TMZCI is given by

TMZCI (A) = 32256m3 n - 129024m3.

Proof: Using Definition 2.4, we have

MZ3 CI(A) = ∏(h∈V(A) [dA (h) τA (h) ]

= ∏_(h∈(E4c)/(E3d ) [dA (h) τA (h)] × ∏(h∈(E6c)/(E4d ) [dA (h) τA (h)] × ∏_(h∈(E7c)/(E4d ) [dA (h) τA (h)]

= |(E4c)/(E3d )| (4×3) × |(E6c)/(E4d )| (6×4) × |(E7c)/(E4d )| (7×4)

= 2m(12) + 2m(24) + m (n-4)(28)

= 32256m3 n - 129024m3.

Theorem 4.9. For a graph A, the FMZCI is given by

FMZCI (A)= 5591040m5 n - 25159660m5.

Proof: Using Definition 2.4, we have

MZ4 CI(A) = ∏(hq∈E(A)  [τA (h) + τA (q)]

= ∏(hq∈E4,4c)  [τA (h) + τA (q)] × ∏(hq∈E4,6c) [τA (h) + τA (q)] × ∏(hq∈E4,7c) [τA (h) + τA (q)] × ∏∏(hq∈E4,3c)  [τA (h) + τA (q)] × ∏∏(hq∈E4,6c)  [τA (h) + τA (q)].

= |E4,4c | (4+4) × |E4,6c |(4+6) × |E6,6c |(6+6) × |E6,7c |(6+7) × |E7,7c |(7+7)

= 2m(8) + 2m(10) + 2m(12) + 2m(13) + (2mn-9m)(14)

= 5591040m5 n - 25159660m5.

Theorem 4.10. For a graph A, the MFMZCI is given by

MFMZCI (A)= 3449830912m5 n - 1642423910m5.

Proof: Using Definition 2.5, we have

MZ1 C* I(A) = ∏(h∈V(A) [dA (h)τA (q) + dA (q)τA (h) ]

= ∏(hq∈(E4,4c)/(E3,3d ) [dA (h)τA (q) + dA (q)τA (h) ] × ∏_(hq∈(E4,6c)/(E3,4d ) [dA (h)τA (q) + dA (q)τA (h) ]

= ∏(hq∈(E6,6c)/(E4,4d ) [dA (h)τA (q) + dA (q)τA (h) ] × ∏(hq∈(E6,7c)/(E4,4d ) [dA (h)τA (q) + dA (q)τA (h) ] × ∏(hq∈(E6,7c)/(E4,4d ) [dA (h)τA (q) + dA (q)τA (h) ]

= |(E4,4c)/(E3,3d )| [ (4×3) + (4×3) ] × |(E4,6c)/(E3,4d )| [ (3×6) + (4×4) ] × |(E6,6c)/(E4,4d )| [ (6×4) + (6×4)] × |(E6,7c)/(E4,4d )| [ (7×4) + (4×6) ] × |(E7,7c)/(E4,4d )| [ (7×4) + (7×4) ]

= 2m(24) × 2m(34) × 2m(48) × 2m(52) × (2mn-9m)(56)

= 3649830912m5 n - 16424239104m5.

Theorem 4.11. For a graph A, the MSMZCI is given by

MSMZCI (A) = 3649830912m5 n - 16424239104m5.

Proof: Using Definition 2.5, we have

MZ2 C* I(A) = ∏(h∈V(A) [dA (h)τA (h) + dA (q)τA (q) ]

= ∏(hq∈(E4,4c)/(E3,3d ) [dA (h)τA (h) + dA (q)τA (q) ] × ∏_(hq∈(E4,6c)/(E3,4d ) [dA (h)τA (h) + dA (q)τA (q) ]

= ∏(hq∈(E6,6c)/(E4,4d ) [dA (h)τA (h) + dA (q)τA (q) ] × ∏(hq∈(E6,7c)/(E4,4d ) [dA (h)τA (h) + dA (q)τA (q) ] × ∏(hq∈(E6,7c)/(E4,4d ) [dA (h)τA (h) + dA (q)τA (q) ]

= |(E4,4c)/(E3,3d )| [ (4×3) + (4×3) ] × |(E4,6c)/(E3,4d )| [ (4×3) + (6×4) ] × |(E6,6c)/(E4,4d )| [ (6×4) + (6×4)] × |(E6,7c)/(E4,4d )| [ (6×4) + (7×4) ] × |(E7,7c)/(E4,4d )| [ (7×4) + (7×4) ]

= 2m(24) × 2m(36) × 2m(48) × 2m(52) × (2mn-9m)(56)

= 3864526848m5 n - 17390370816m5.

Theorem 4.12. For a graph A, the MTMZCI is given by

MTMZCI (A) = 4.02728883 × 1014 m5 n - 1.81227997 × 1015 " " m5.

Proof: Using Definition 2.5, we have

MZ3 C* I(A) = ∏(h∈V(A) [dA (h)τA (h) × dA (q)τA (q) ]

= ∏(hq∈(E4,4c)/(E3,3d ) [dA (h)τA (h) × dA (q)τA (q) ] × ∏_(hq∈(E4,6c)/(E3,4d ) [dA (h)τA (h) × dA (q)τA (q) ]

= ∏(hq∈(E6,6c)/(E4,4d ) [dA (h)τA (h) × dA (q)τA (q) ] × ∏(hq∈(E6,7c)/(E4,4d ) [dA (h)τA (h) × dA (q)τA (q) ] × ∏(hq∈(E6,7c)/(E4,4d ) [dA (h)τA (h) × dA (q)τA (q) ]

= |(E4,4c)/(E3,3d )| [ (4×3) × (4×3) ] × |(E4,6c)/(E3,4d )| [ (4×3) × (6×4) ] × |(E6,6c)/(E4,4d )| [ (6×4) × (6×4)] × |(E6,7c)/(E4,4d )| [ (6×4) × (7×4) ] × |(E7,7c)/(E4,4d )| [ (7×4) × (7×4) ]

= 2m(24) × 2m(36) × 2m(48) × 2m(52) × (2mn-9m) (56)

= 4.02728883 × 1014 m5 n - 1.81227997 × 1015 m5.

5. COMPARATIVE ANALYSIS

In this section, we compare all the computed values of connection based ZIs with each other. The numerical results of the Cartesian product of path graphs with cycle graphs on the basis of connection based ZIs such as FZCI, SZCI, MFZCI, MSZCI, MTZCI, FMZCI, SMZCI, TMZCI, FMZCI, MFMZCI, MSMZCI, and MTMZCI are presented in Table 5 and their graphical representation is shown in Figure 3.

Table 5.Computed Values of FZCI, SZCI, MFZCI, MSZCI, MTZCI, FMZCI, SMZCI, TMZCI, FMZCI, MFMZCI, MSMZCI, and MTMZCI of Graph A for m,n =01,02,03,⋯08

ZCIs m=1,n=1 m=2,n=2 m=3,n=3 m=4,n=4 m=5,n=5 m=6,n=6 m=7,n=7
Z1 CI(A

40

187

432

775

1216

1755

2392
Z2 CI(A)

-107

187

677

1362

2245

1559

4597
Z1 CI* (A)

-12

72

252

408

968

1185

1332
Z2 C* I(A)

-76

260

820

1604

2612

3844

5300
Z3 C* I(A)

-72

256

816

1600

2608

3840

5296
MZ1 CI(A)

-338688

-1806336

-3048192

0

14112000

48771072

116169984
MZ2 CI(A)

-4087480356

-1.3050

-9.8874

- 4.1570

-1.1350

-3.3142

-6.7761
MZ3 CI(A)

-96768

-51609

-870912

-2064384

4032000

13934592

33191424
MZ4 CI(A)

-19568640

-447283200

-2037934080

-2862612480

873600000

6.5213

2.3492
MZ4 CI(A)

-1.6059

-2.9198

-1.6954

-1.8687

-5.1952

4.2571

1.5335
MZ2 CI* (A)

-1.3525

-3.0916

-1.4086

1.9786

6.0383

4.5075

6.5335
MZ3 CI* (A)

-1.4095

-3.2218

-1.4679

-2.0619

6.2926

4.6974

1.6921


Figure 3. Computed Values of ZCIs of Graph A for m,n=1,2,3,⋯08

By examining both Table 5 and Figure 3, it becomes evident that the Cartesian product of path graphs, cycle graphs. In Figure 3 show that FZCI has consistently achieved the highest values within this network. Graphical representation in Figure 5 illustrates that FZCI has a higher line than all the other ZCIs within the Cartesian product of path graphs and cycle graphs. The computed results are universal and contingent solely on the values of m,n..

6. CONCLUSION

In this research, we have derived expressions to compute TIs for the Cartesian product of cycle graphs and path graphs. TIs are crucial in manipulating and analyzing chemical organizational information. The study encompasses the calculation of various TIs, such as the first Zagreb connection index (FZCI) and the second Zagreb connection index (SZCI). Additionally, we have computed MFZCI, MSZCI, and MTZCI, as well as FMZCI, SMZCI, TMZCI, and FMZCI. A comprehensive comparative analysis of all the computed TIs is presented, leading to the conclusion that MFMZCI, MSMZCI, and MTMZCI exhibit greater efficacy in predicting the physicochemical properties of the chemical network. This mathematical investigation not only simplifies the understanding of the chosen structure but also serves as a motivation to delve into the study of organic networks. In future, we are interested to compute the ZCIs of other graphs, including prism graphs, line graphs etc.

Conflict of Interest

The author of the manuscript has no financial or non-financial conflict of interest in the subject matter or materials discussed in this manuscript.

Data Availability Statement

The data associated with this study will be provided by the corresponding author upon request.

Funding Details

No funding has been received for this research.

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