Novel/Old Generalized Multiplicative Zagreb Indices of Some Special Graphs

1. INTRODUCTION

Recently, graph theory has provided some useful tools for the study of different structures. These tools are known by the name of topological indices (TIs) and used mostly in pharmaceutical industry, medicinal fields, and the study of crystalline and nano materials, see [1-3]. Additionally, TIs are used in quantitative-structure-activity relationships (QSARs) and quantitative-structure-property relationships (QSPRs). These relations have been joined to molecular-structures with their biological-structures [4, 5]. Mathematically, TIs can be defined as where is a real-valued function, is a molecular-structure, and is a real value which depends on .

Multiplicative Zagreb indices (ZIs) have outstanding importance in varius research fields. They play a significant role in analytical chemistry, toxicology, medicine, physical chemistry, pharmaceutical research, material sciences, environmental chemistry, and engineering. They can be joinedwith several properties of chemical compounds [6]. Todeshine et al. [7] and Eliasi et al. [8] separately developed multiplicative ZIs. In 2011, Gutman [9] used multiplicative ZIs to compute results of different tree graphs. Xu and Hua [10] used these multiplicative versions with the name of multiplicative sum ZI to find different results of trees, as well as unicyclic and bicyclic graphs. Xu and Das [11] also used multiplicative sum ZI to find different results of trees, as well as unicyclic and bicyclic graphs, by using another method shorter than [8]. Multiplicative ZIs and multiplicative sum ZIs are being rapidly used in the forthcoming research. For more information, see [12–18].

In 2011, Azari [19] defined the generalized ZI to compute nanotubes and nanotori networks. He also defined graphical products, such as Cartesian product, sum, lexicographic product, union, corona product, disjunction, strong product, and symmetric difference with the help of generalized ZI [20]. Farahani and Kanna [21] computed generalized ZI for V-phenylenic nanotori and nanotubes. Sarkar et al. [22] studied the generalized ZI of several allotropes of carbon, such as carbon graphite, crystal cubic structure of carbon, and graphene. Sarkar et al. [23] used the generalized ZI findings to compute different results of regular dendrimers. Sarkar et al. [24] used this generalized ZI with another name, that is, -Zagreb index to compute several derived networks. Mirajkar et al. [25] computed the generalized ZI of capra operation of cycle C_v on v-vertces. Wang et al. [26] used both generalized and generalization ZIs to compute several results of silicon-carbon graphs.

Degree-based TIs have been classified into degree and connection number (CN) [27]. Degree and CN indicate the number of vertices whose distance from vertex must be 1 and 2, respectively. These indices have a powerful role in the study of chemical compounds and biological experiments [28, 29]. The first degree-based TI came into existence from the study of -electron energy. This TI was revolutionized by Gutman and Trinajsti [30] with the name of ZI. They also shortly worked on another TI named as CN TI which is the number of those vertices whose distance from a particular vertex is 2. As such, TIs are based on CN [31]. Currently, these CN-based ZIs are used to investigate the physicochemical properties of chemical compounds, such as their stability, boiling point, strain energy, acentric factor, and entropy more than classical ZIs, see [32–34]. Ye et al. [35] computed ZCIs for nanotubes. Wang et al. [36] computed ZCIs of -dimensional Benes networks. Hussain et al. [37] computed ZCIs of subdivided graph on . For more details related to CN-based ZIs or leap ZIs, the reader may see [38–47].

In this paper, the generalized first, second, third, and fourth multiplicative ZCIs are defined. Exact solutions for the old and novel generalized multiplicative ZIs of some special graphs including wheel, gear, helm, flower, and sunflower are also developed. The rest of the paper is structured as follows: Section II gives the basic definitions of the degree- and CN-based ZIs, Section III provides the definitions of several special graphs and their main results which are related to generalized multiplicative ZIs and generalized multiplicative ZCIs, while section IV presents the conclusion.

2. NOTATIONS AND PRELIMINARIES

Let 𝑄 be a connected and simple graph. |E(Q)|"=s" and |V(Q)|"=s" are the size and order of graph 𝑄, respectively. The degree (𝑑_Q.(𝑚)) of vertex 𝑚 is the number of incident edges. If we sum the degrees of all the vertices adjacent to 𝑚 it is known as degree sum of a vertex. Mathematically it is written as DS(m)=∑_{(l∈NQ (m))}d_Q(l). If we multiply the degrees of all the vertices adjacent to 𝑚 it is known as degree product of a vertex. Mathematically it is written as DP(m)=∏_{(l∈N_Q (m))}d_Q(l). If 𝑁_𝛿(𝑚) is the 𝛿 -neighborhood of a vetex 𝑚 then 𝑁_𝛿(𝑚)=d_𝛿(𝑚). (number of 𝛿 -neighbors of a vertex 𝑚). If we put 𝛿=2 it yields 𝑑-2.(𝑚)=𝜏(𝑚) (CN of a vertex 𝑚). Furthermore 𝑑_Q (𝑚)=∑_{𝑚∈𝑉(𝑄)}-𝑑_Q(𝑚)/𝑣 and 𝜏_Q (𝑚)=∑_{𝑚∈𝑉(𝑄)}-𝜏_Q(𝑚)/𝑣 are the average degree and average CN of order 𝑣.The computed value of and are and , respectively. For more terminologies and notions (d_Q)(m)(τ_Q)(m)2s/v(M₁-2s)/v.[48]

Definition 2.1 [42] Let Q be a connected graph and ∀κ,λ∈R-{0}. The δ -distance reneralized Zagreb index (Z_(κ,λ)^δ (Q)) is

Z_(κ,λ)^δ (Q)=∑_{(lm ∈E(Q)})[d_δ (l)^κ d_δ (m)^λ +d _δ (m)^κ d_δ (l)^λ].

where δ≥1. If we put δ=1 , we get the generalized ZI as follows:

Z_(κ,λ) (Q)=∑_{(lm ∈E(Q)})[d(l)^κ d(m)^λ+d(m)^κ d(l)^λ].

Azari and Iranmanesh [19] (2011) defined this degree-based generalized version. If we replace sum by product, we obtain the generalized multiplicative ZI as follows:

MZ_(κ,λ) (Q)=∏_{(lm ∈E(Q)})[d(l)^κ d(m)^λ+ d(m)^κ d(l)^λ].

If we put δ=2 , we get generalized ZCIs based on connection number (τ=d ₂ ) as follows:

Definition 2.2 [42] Let Q be a connected graph and ∀κ,λ∈R-{0}. The generalized first, second, third, and fourth ZCIs are as follows:

C_(κ,λ) (Q)=∑_(lm ∈E(Q))[τ(l)^κ τ(m)^λ +τ(m)^κ τ(l)^λ]

C_(κ,λ)² (Q)=∑_(lm ∈E(Q))[d(l)^κ τ(m)^λ +d(m)^κ τ(l)^λ]

C_(κ,λ)³ (Q)=∑_(lm ∈E(Q))[d(l)^κ τ(l)^λ +d(m)^κ τ(m)^λ]

C_(κ,λ)⁴ (Q)=∑_(lm ∈E(Q))[d(l)^κ τ(l)^λ ×d(m)^κ τ(m)^λ].

If we change sum into product, we get generalized multiplicative ZCIs as follow

Definition 2.3 Let Q be a connected graph and ∀κ,λ∈R-{0}. The generalized first, second, third, and fourth multiplicative ZCIs are as follows:

MC_(κ,λ)(Q)=∏_(lm∈E(Q))[τ(l)^κ τ(m)^λ+τ(m)^κ τ(l)^λ]

MC_(κ,λ)² (Q)=∏_(lm∈E(Q))[d(l)^κ τ(m)^λ+d(m)^κ τ(l)^λ]

MC_(κ,λ)³ (Q)=∏_(lm∈E(Q))[d(l)^κ τ(l)^λ+d(m)^κ τ(m)^λ]

MC_(κ,λ)⁴ (Q)=∏_(lm∈E(Q))[d(l)^κ τ(l)^λ×d(m)^κ τ(m)^λ]

3. DEFINITIONS AND MAIN RESULTS

This section presents the definitons of some special graphs, such as wheel, gear, helm, flower, and sunflower graphs, respectively. Also, main results for the generalized multiplicative ZI and generalized multiplicative ZCIs are presented.

The wheel graph W_v is defined by joining K_1 and C_v, where K_1 and C_v are the complete and cyclic graphs of orders 1 and v respectively. Thus, v"+1" and 2v are the order (|V(W_v )|) and (|E(W_v )|) of wheel graph, respectively. Apex is the vertex corresponding to K_1 and rim is the set of the vertices corresponding to C_v. Figure 1 depicts the graphical representation of wheel graph.

Figure 1. Wheel Graph W_v

Table 1. Partition of Vertices for

Theorem 3.1. Let W_v be a wheel graph with order v"+1" . The generalized multiplicative ZI and the generalized first, second, third, and fourth multiplicative ZCIs of W_vare as follows:

1. MZ_(κ,λ) (W_v) = 2 ^r 3^(κ+λ)v×(r^κ 3^λ + 3 ^κ r^λ )^v,

2. MC_(κ,λ) (W_v ) =0,

3. MC_(κ,λ)² (W_v) = 2 ^v (3v)^κv (v -3 )²λv

4. MC_(κ,λ)³ (W_v) = 2 ^v (9)^κv (v -3 )²λv

5. MC_(κ,λ)⁴ (W_v ) =0.

Proof: 1 Since W_v is a wheel graph of order v +1 , where v≥4. Also, m₀ is the apex vertex and {m_1,m₂,……,m_v } is the set of rim vertices for W_v. Then, by definition,

∏_(lm∈E(Wv)) [d^(W_v) ) (l)^κ d^(W_v) ) (m)^λ+d^(W_v) ) (m)^κ d^(W_v) ) (l)^λ ]

∏_(m0) m_i∈E(W_v )) [d^(W_v) ) (m₀ )^κ d^(W_v) ) (m_i )^λ+d^(W_v) ) (m_i )^κ d_

>(W_v ) (m₀ )^λ ]× ∏_(mi) m_i +1 ∈E(W_v )) [d^(W_v) ) (m_i )^κ d^(W_v) )

>(m_i +1 )^λ+d^(W_v) ) (m_i +1 )^κ d^(W_v) ) (m_i )^λ ].

By using Table 1, we get

=[v^κ 3^λ + 3 ^κ v^λ ]^v× [3^κ 3^λ + 3 ^κ 3^λ ]^v

= 2 ^v 3^(κ+λ)v × (v^κ 3^λ + 3 ^κ v^λ )^v.

2.MC_(κ,λ) (W_v ) ∏_(lm∈E(Wv)) [τ^(W_v) ) (l)^κ τ^(W_v) ) (m)^λ+τ^(W_v) )

>(m)^κ τ^(W_v) ) (l)^λ ]

∏_(m0) m_i∈E(W_v )) [τ^(W_v) ) (m₀ )^κ τ^(W_v) ) (m_i )^λ+τ^(W_v) ) (m_i )^κ

τ^(W_v) ) (m₀ )^λ ] × ∏_(mi) m_i +1 ∈E(W_v )) [τ^(W_v) ) (m_i )^κ τ_

=[0^κ (v-3)^λ+( v-3 )^κ 0^λ ]^v × [( v-3 )^κ (v-3)^λ+( v-3 )^κ (v-3)^λ ]^v

=0 .

3. MC_(κ,λ)² (W_v ) ∏_(lm∈E(Wv)) [d^(W_v) ) (l)^κ τ^(W_v) ) (m)^λ+d_

∏_(m0) m_i∈E(W_v )) [d^(W_v) ) (m₀ )^κ τ^(W_v) ) (m_i )^λ+d^(W_v) ) (m_i )

^κ τ^(W_v) ) (m₀ )^λ ] × ∏_(mi) m_i +1 ∈E(W_v )) [d^(W_v) ) (m_i )

=[r^κ (r-3)^λ + 3 ^κ 0^λ ]^r × [3^κ (r-3)^λ + 3 ^κ (r-3)^λ ]^r

= 2 ^v (3v)^κv (v-3)²λv.

4. MC_(κ,λ)³ (W_v ) ∏_(lm∈E(Wv)) [d^(W_v) ) (l)^κ τ^(W_v) ) (l)

∏_(m0) m_i∈E(W_v )) [d^(W_v) ) (m₀ )^κ τ^(W_v) ) (m₀ )^λ+d_

=[v^κ 0^λ + 3 ^κ (v-3)^λ ]^v × [3^κ (v-3)^λ + 3 ^κ (v-3)^λ ]^v

= 2 ^v (9)^κv (v-3)²λv.

5. MC_(α,β)⁴ (W_v ) ∏_(lm∈E(Wv)) [d^(W_v) ) (l)^κτ_

∏_(m0) m_i∈E(W_v )) [d^(W_v)) (m₀ )^κ τ_(Wv) (m₀ )^λ×d^(W_v) ) (m_i )^κ τ^(W_v) ) (m_i )^λ ] ×

∏_(mi) m_i+1 ∈E(W_v )) [d^(W_v) ) (m_i )^κ τ^(W_v) ) (m_i )^λ×d^(W_v) ) (m_i+1 )^κ τ^(W_v) ) (m_i+1 )^λ ]

=[v^κ 0^λ×3^κ (v-3)^λ ]^v × [3^κ (v-3)^λ×3^κ (v-3)^λ ]^v

=0 .

3.2. Gear Graph G_v

The gear graph G_v is obtained from the wheel graph by including a new vertex between each pair of the adjacent vertices of rim. Thus, 2v+1 and 3v are the order (|V(G_v )|) and size (|E(G_v )|) of gear graph, respectively. Also, the bipartite wheel graph is called gear graph. Figure 2 depicts the graphical representation of gear graph.

Figure 2. Wheel Graph G_v

Table 2. Partition of Vertices for G_v

Theorem 3.2. Let G_v be a gear graph with order 2v+1 .

The generalized multiplicative ZI and the generalized first, second, third, and fourth multiplicative ZCIs of G_v are as follows:

1. MZ_(κ,λ) (G_v)=[v^κ 3^λ + 3 ^κ v^λ ]^v×[3^κ 2^λ + 2 ^κ 3^λ )²v

2. MC_{(κ λ)} (G_v )=[v^κ (v-1)^λ+(v-1 )^κ v^λ ]^v × [(v-1)^κ 3^λ + 3 ^κ (v-1)^λ ]²v

3. MC_(κ,λ)² (G_v )=[v^κ (v-1)^λ + 3 ^κ v^λ ]^r × [3^κ 3^λ + 2 ^κ (v-1)^λ ]²v

4. MC_(κ,λ)³ (G_v )=[v^κ v^λ + 3 ^κ (v-1)^λ ]^v× [3^κ (v-1)^λ + 2 ^κ 3^λ ]²v

5. MC_(κ,λ)⁴ (G_v)= 2 ²κv×3^(3κ+2 λ)v×r^(κ+λ)v×(r-1)³λv.

Proof: 1. Since G_v is a gear graph of order 2v+1 . Also, m₀ is the apex vertex; {m_1,m₂,……,m_v } and {l_1,l₂,……,l_v } are the sets of rim vertices for G_v. Then, by definition,

∏_(lm∈E(G_v )) [d_(G_v ) (l)^κ d_(G_v ) (m)^λ+d_(G_v ) (m)^κ d_(G_v ) (l)^λ ]

∏_(m0) m_i∈E(G_v )) [d_(G_v ) (m₀ )^κ d_(G_v ) (m_i )^λ+d_(G_v ) (m_i )^κ

d_(G_v ) (m₀ )^λ ]× ∏_(mi) l_i∈E(G_v )) [d_(G_v ) (m_i )^κ d_(G_v ) (l_i )

^λ+d_(G_v ) (l_i )^κ d_(G_v ) (m_i )^λ ].

By using Table 2, we get

=[v^κ 3^λ + 3 ^κ v^λ ]^v× [3^κ 2^λ + 2 ^κ 3^λ ]²v.

2. MC_(κ,λ) (G_v ) ∏_(lm∈E(G_v )) [τ_(G_v ) (l)^κ τ_(G_v ) (m)^λ+τ_(G_v ) (m)^κ τ_(G_v ) (l)^λ ]

∏_(m0) m_i∈E(G_v )) [τ_(G_v ) (m₀ )^κ τ_(G_v ) (m_i )^λ+τ_(G_v ) (m_i )^κ τ

(G_v ) (m₀ )^λ ] × ∏_(mi) l_i∈E(G_v )) [τ_(G_v ) (m_i )^κ τ_(G_v ) (l_i )^λ+τ_(G_v ) (l_i )^κ τ_(G_v ) (m_i )^λ ]

=[v^κ (v-1)^λ+(v-1 )^κ v^λ ]^r × [(v-1)^κ 3^λ + 3 ^κ (v-1)^λ ]²v.

3. MC_(κ,λ)² (G_v ) ∏_(lm∈E(G_v )) [d_(G_v ) (l)^κ τ_(G_v ) (m)^λ+d_(G_v ) (m)^κ τ_(G_v ) (l)^λ ]

∏_(m0) m_i∈E(G_v )) [d_(G_v ) (m₀ )^κ τ_(G_v ) (m_i )^λ+d_(G_v ) (m_i )^κ τ_

(G_v ) (m₀ )^λ ] × ∏_(mi) l_i∈E(G_v )) [d_(G_v ) (m_i )^κ τ_(G_v ) (l_i )^λ+d_(G_v ) (l_i )^κ τ_(G_v ) (m_i )^λ ]

=[v^κ (v-1)^λ + 3 ^κ v^λ ]^v × [3^κ 3^λ + 2 ^κ (v-1)^λ ]²v.

4. MC_(κ,λ)³ (G_v ) ∏_(lm∈E(G_v )) [d_(G_v ) (l)^κ τ_(G_v ) (l)^λ+τ_(G_v ) (m)^κ d_(G_v ) (m)^λ ]

∏_(m0) m_i∈E(G_v )) [d_(G_v ) (m₀ )^κ τ_(G_v ) (m₀ )^λ+d_(G_v ) (m_i )^κ p>τ_

(G_v ) (m_i )^λ ] × ∏_(mi) l_i∈E(G_v )) [d_(G_v ) (m_i )^κ τ_(G_v )

(m_i )^λ+d_(G_v ) (l_i )^κ τ_(G_v ) (l_i )^λ ]

=[v^κ v^λ + 3 ^κ (v-1)^λ ]^v× [3^κ (v-1)^λ + 2 ^κ 3^λ ]²v.

5. MC_(κ,λ)⁴ (G_v ) ∏_(lm∈E(G_v )) [d_(G_v ) (l)^κ τ_(G_v ) (l)^λ×τ_(G_v ) (m)^κ d_(G_v ) (m)^λ ]

∏_(m0) m_i∈E(G_v )) [d_(G_v ) (m₀ )^κ τ_(G_v ) (m₀ )^λ×d_(G_v ) (m_i )^κ

τ_(G_v ) (m_i )^λ ] × ∏_(mi) l_i∈E(G_v )) [d_(G_v ) (m_i )^κ τ_(G_v )

(m_i )^λ×d_(G_v ) (l_i )^κ τ_(G_v ) (l_i )^λ ]

=[v^κ v^λ×3^κ (v-1)^λ ]^v × [3^κ (v-1)^λ×2^κ 3^λ ]²v

= 2 ²αv×3^(3α+2 β)v×r^(α+β)v×(r-1)³λv.

3.3. Helm Graph H_v

The helm graph H_vis obtained from the wheel graph by joining a pendant edge to every vertex of the rim. Thus, 2v+1 and 3v are the order (|V(H_v )|) and size (|E(H_v )|) of helm graph, respectively. Figure 3 depicts the graphical representation of helm graph.

Theorem 3.3. LetH_v be a helm graph with order 2v+1 . The generalized multiplicative ZI and the generalized first, second, third, and fourth multiplicative ZCIs of H_v are as follows:

1. MZ_(κ,λ) (H_v)= 2 ^v 4^(κ+λ)v×(4^κ + 4 ^λ )^v×[r^κ 4^λ + 4 ^κ r^λ ]^v,

2. MC_(κ,λ) (H_v)= 2 ^v×(v-1)^(κ+λ)v×[v^κ (v-1)^λ+(v-1 )^κ v^λ ]^r×[(v-1)^κ 3^λ + 3 ^κ (v-1)^λ ]^v

Figure 3.Helm Graph H_v

Table 3. Partition of Vertices for H_v

4. CONCLUSION

Topological indices (TIs) are the mathematical coding of molecular graphs that predict the physicochemical, toxicological, biological, and structural properties of chemical compounds that are directly linked with these graphs. The Zagreb connection indices (ZCIs) are among the TIs of molecular graphs that depend upon the connection number (CN). These CN-based TIs are well used in the study of quantitative structures activity relationships (QSARs) and quantitative structures property relationships (QSPRs). These days, CN-based multiplicative Zagreb indices are the best tools available for the study of QSARs and QSPRs. In this paper, the generalized first, second, third, and fourth multiplicative ZCIs were defined. Also, some exact solutions of the novel/old generalized multiplicative Zagreb indices for some special graphs, namely flower, sunflower, wheel, helm, and gear graphs were computed.

CONFLICT OF INTEREST

The authors of the manuscript have no financial or non-financial conflict of interest in thesubject matter or materials discussed in this manuscript

DATA AVALIABILITY STATEMENT

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