On the first and second Zagreb indices of some products of signed graphs

Abstract

Some of the most comprehensively studied degree-based topological indices are the Zagreb indices. In this article, the pair of Zagreb indices have been determined for five product graphs namely tensor product, Cartesian product, lexicographic product, strong product, symmetric difference of G₁ and G₂ in terms of the Zagreb indices of the signed graphs of the factor graphs G₁ and G₂ and their orders and sizes.

KEYWORDS:

• Signed graph
• Zagreb index
• tensor product
• Cartesian product
• lexicographic product
• strong product
• symmetric difference

1 Introduction

A graph $G=G(V,E)$ consists of a nonempty set V and a collection E of unordered pairs of distinct elements from V. The elements of V and E are called vertices and edges of G, respectively. The number of edges incident on a vertex u is called the degree of u in G and it is denoted by $d_G(u)$. For, other basic concepts and terminologies on graphs [9] and [3] can be referred to. The graphs considered in this study are connected and simple i.e., without loops and multiple edges. When the edges of a graph are assigned a + or – sign then the graph obtained is called signed graph. Formally, a signed graph is a graph G(V, E) along with a function $\mu :E(G)\to \{+,-\}$ that assigns a sign, either + or –, to every edge in G. It is denoted by $S(G,\mu )$. The function μ is called the signature function or sign function of S. The unsigned graph G is called the underlying graph of the signed graph S. The number of positive edges incident on a vertex u is the positive degree of u, denoted by $d_S^{+}(u)$ and the number of negative edges incident on a vertex u is the negative degree of u and is denoted by $d_S^{-}(u)$. Clearly, $d_G(u)=d_S^{+}(u)+d_S^{-}(u)$. When there is no confusion about the graph under consideration, we use $d_u,d_u^{+},d_u^{-}$ to mean degree, positive degree and negative degree respectively.

Graph theory is widely used in the characterization of structures by means of numerical parameters known as topological indices. The topological indices, as their name suggests are highly correlated with the topology (i.e. the arrangement of nodes and links between them) of a network, and give an insight into its properties. Ever since, the inception of Weiner index W(G) in 1946 by Alfred Weiner, there has been a huge explosion over the years in the development of topological indices based on various parameters of a graph G. For more information the following works can be referred to and the references cited therein [10, 12–14]. In 1972, Gutman and Trinajstic, who were members of the “Rudjer Boskovic” Institute in Zagreb, Croatia introduced a pair of topological indices. These indices thus came to be known by various names like the Zagreb group indices, the Zagreb group parameters and, mostly the Zagreb indices, denoted by symbols $M_1(G)$ and $M_2(G)$. They inferred that the dependence of the total π-electron energy on molecular structure is strongly correlated with these indices [5]. Several mathematical properties and chemical applications of these two indices have been reported in [6–8]. Graph operations play a very important role in determining the relation between the product graph and its constituent graphs. The general expressions of the indices of graph operations thus deduced enhance the understanding of relations between the values of topological indices of the product graphs and those of their corresponding components. More study on topological indices related to product graphs is provided in [1, 2, 4]. In 2009, these indices were computed for graph operations like the Cartesian product, composition, join, symmetric difference and disjunction [11]. In 2020, the first and the second Zagreb indices for signed graphs was studied in [15]. Taking motivation from these two works, in this present paper the work on Zagreb indices has been extended to signed graphs. The objective of this article is to study the mathematical properties of these pair of indices for various graph operations. However, their applications in terms of chemical graph theory are not discussed here.

1.1 Preliminaries

The results in this paper are obtained using the definitions given in Tables 1 and 2.

Table 1 Some indices, their notations and definitions.

Name of the index H	Notation	Definition
The first Zagreb index of G	$M_1(G)$	$\mathrm{(}\text{Tex translation failed}\mathrm{)}$.
The second Zagreb index of G	$M_2(G)$	$M_2(G)=\sum _{\{u,v\}\in E(G)}{d}_G(u)d_G(v)$.
The first positive Zagreb index of S	$M_1^{+}(S)$	$M_1^{+}(S)=\sum _{v_i\in V(G)}{(d_{v_i}^{+})}^2$.
The first negative Zagreb index of S	$M_1^{-}(S)$	$M_1^{-}(S)=\sum _{v_i\in V(G)}{(d_{v_i}^{-})}^2$.
The first mixed Zagreb index of S	$M_1^{*}(S)$	$M_1^{*}(S)=\sum _{v_i\in V(G)}(d_{v_i}^{+})(d_{v_i}^{-}).$
The second positive Zagreb index of S	$M_2^{+}(S)$	$M_2^{+}(S)=\sum _{v_i{v}_j\in E(G)}{d}_{v_i}^{+}{d}_{v_j}^{+};1\le i\ne j\le n$.
The second negative Zagreb index of S	$M_2^{-}(S)$	$\sum _{v_i{v}_j\in E(G)}{d}_{v_i}^{-}{d}_{v_j}^{-};1\le i\ne j\le n$.
The second mixed Zagreb index of S	$M_2^{*}(S)$	$\sum _{v_i{v}_j\in E(G)}{d}_{v_i}^{+}{d}_{v_j}^{-};1\le i\ne j\le n$.

Table 2 Adjacency rules and degree of every vertex (u₁, u₂) in some graph products.

Product graph H	Notation	Adjacency rule	Degree of $(u_1,u_2)\in V(H)$
Tensor product	$G_1\otimes G_2$	$(u_1,v_1)\sim (u_2,v_2)$ iff $u_1{u}_2\in E(G_1)$ and $v_1{v}_2\in E(G_2)$.	$d_{(G_1\otimes G_2)}(u_1,u_2)=d_{G_1}(u_1)d_{G_2}(u_2).$
Cartesian product	$G_1\square G_2$	$(u_1,u_2)\sim (v_1,v_2)$ iff $u_1{v}_1\in E(G_1)$ and $u_2=v_2$ or u1  =  v1 and $u_2{v}_2\in E(G_2)$.	$d_{(G_1\square G_2)}(u_1,u_2)=d_{G_1}(u_1)+d_{G_2}(u_2).$
Lexicographic product or composition	$G_1[G_2]$	$(u_1,u_2)\sim (v_1,v_2)$ iff $u_1{v}_1\in E(G_1)$ or u1  =  v1 and $u_2{v}_2\in E(G_2)$.	$d_{(G_1[G_2])}(u_1,u_2)=n_2{d}_{G_1}(u_1)+d_{G_2}(u_2).$
Strong product	$G_1*G_2$	$(u_1,u_2)\sim (v_1,v_2)$ iff $u_1{v}_1\in E(G_1)$ and $u_2=v_2$ or u1  =  v1 and $u_2{v}_2\in E(G_2)$ or $u_1{v}_1\in E(G_1)$ and $u_2{v}_2\in E(G_2)$.	$d_{(G_1*G_2)}(u_1,u_2)=d_{G_1}(u_1)+d_{G_2}(u_2)+d_{G_1}(u_1)d_{G_2}(u_2).$
Symmetric difference	$G_1\oplus G_2$	$(u_1,u_2)\sim (v_1,v_2)$ iff $u_1{v}_1\in E(G_1)$ or $u_2{v}_2\in E(G_2)$ but not both.	$d_{(G_1\oplus G_2)}(u_1,u_2)=n_2{d}_{G_1}(u_1)+n_1{d}_{G_2}(u_2)-2d_{G_1}(u_1)d_{G_2}(u_2).$

Lemma 1.1.

[15] For a signed graph S and its underlying graph G, (1) $$M_1(G)=M_1^{+}(S)+M_1^{-}(S)+2M_1^{*}(S)$$(1) (2) $$M_2(G)=M_2^{+}(S)+M_2^{-}(S)+M_2^{*}(S)$$(2)

Lemma 1.2.

[11] Let G₁ and G₂ be two graphs. Then (3) $$|E(G_1\otimes G_2)|=2|E(G_1)||E(G_2)|.$$(3) (4) $$|E(G_1\square G_2)|=|E(G_1)||V(G_2)|+|V(G_1)||E(G_2)|.$$(4) (5) $$|E(G_1\left[G_2\right])|=|E(G_1)||V(G_2){|}^2+|E(G_2)\left|\right|V(G_1)|.$$(5) (6) $$\begin{array}{c}|E(G_1*G_2)|=|E(G_1)||V(G_2)|+|E(G_2)||V(G_1)|\\ +2|E(G_1)||E(G_2)|.\end{array}$$(6) (7) $$\begin{array}{c}|E(G_1\oplus G_2)|=|E(G_1)||V(G_2){|}^2+|E(G_2)\left|\right|V(G_1){|}^2\\ -4|E(G_1)||E(G_2)|.\end{array}$$(7)

2 Main results

Theorem 2.1.

Let $(S_1,{\sigma }_1)$ and $(S_2,{\sigma }_2)$ be two signed graphs and their underlying graphs be G₁ and G₂, respectively. Then the first Zagreb index of the tensor product of G₁ and G₂, in terms of the Zagreb index of their signed graphs is given by: (8) $$M_1(G_1\otimes G_2)=M_1(S_1)M_1(S_2)$$(8) where $M_1(S_1)=[M_1^{+}(S_1)+M_1^{-}(S_1)+2M_1^{*}(S_1)]$ and $M_1(S_2)=[M_1^{+}(S_2)+M_1^{-}(S_2)+2M_1^{*}(S_2)]$.

Proof.

By applying the results in the Tables 1 and 2 we obtain, $$\begin{array}{ccc}{M}_1(G_1\otimes G_2)\hfill & =\hfill & \mathrm{}\sum _{(u_i,v_j)\in V(G_1\otimes G_2)}{d}_{(u_i,v_j)}^2=\mathrm{}\sum _{(u_i,v_j)\in V(G_1\otimes G_2)}{(d_{u_i}{d}_{v_j})}^2\hfill \\ \hfill & =\hfill & \sum _{u_i\in V(G_1)}{(d_{u_i})}^2{\sum }_{v_j\in V(G_2}{(d_{v_j})}^2\hfill \\ \hfill & =\hfill & \sum _{u_i\in V(G_1)}{(d_{u_i}^{+}+d_{u_i}^{-})}^2\sum _{v_j\in V(G_2)}{(d_{v_j}^{+}+d_{v_j}^{-})}^2\hfill \\ \hfill & =\hfill & \sum _{u_i\in V(G_1)}(d_{u_i}^{+}{}^2+d_{u_i}^{-}{}^2+2d_{u_i}^{+}{d}_{u_i}^{-})\hfill \\ \hfill & \hfill & \sum _{v_j\in V(G_2)}(d_{v_j}^{+}{}^2+d_{v_j}^{-}{}^2+2d_{v_j}^{+}{d}_{v_j}^{-}),\hfill \end{array}$$which gives us Equation (8)(8) $$M_1(G_1\otimes G_2)=M_1(S_1)M_1(S_2)$$(8) . □

Theorem 2.2.

Let $S_1(G_1,{\sigma }_1)$ and $S_2(G_2,{\sigma }_2)$ be two signed graphs, with n₁, n₂ and m₁, m₂ being the number of vertices and number of edges of G₁ and G₂, respectively. Then the first Zagreb index of the Cartesian product of G₁ and G₂, in terms of the Zagreb index of their signed graphs is given by (9) $$M_1(G_1\square G_2)=n_2{M}_1(S_1)+n_1{M}_1(S_2)+8m_1{m}_2$$(9) where $M_1(S_1)=[M_1^{+}(S_1)+M_1^{-}(S_1)+2M_1^{*}(S_1)]$ and $M_1(S_2)=[M_1^{+}(S_2)+M_1^{-}(S_2)+2M_1^{*}(S_2)]$.

Proof.

By applying the results in the Tables 1 and 2 we obtain, $$\begin{array}{ccc}{M}_1(G_1\square G_2)\hfill & =\hfill & \sum _{(u_i,v_j)\in V(G_1\square G_2)}{\left[d_{(u_i,v_j)}\right]}^2\hfill \\ \hfill & =\hfill & \sum _{(u_i,v_j)\in V(G_1\square G_2)}{(d_{u_i}+d_{v_j})}^2\hfill \\ \hfill & =\hfill & \sum _{(u_i,v_j)\in V(G_1\square G_2)}[{(d_{u_i})}^2+{(d_{v_j})}^2+2(d_{u_i}{d}_{v_j})]\hfill \\ \hfill & =\hfill & \mathrm{}\sum _{u_i\in V(G_1)}{(d_{u_i})}^2\mathrm{}\sum _{v_j\in V(G_2)}1+\mathrm{}\sum _{v_j\in V(G_2)}{(d_{v_j})}^2\mathrm{}\sum _{u_i\in V(G_1)}1\hfill \\ \hfill & \hfill & +2\sum _{u_i\in V(G_1)}{d}_{u_i}\sum _{v_j\in V(G_2)}{d}_{v_j}\hfill \\ \hfill & =\hfill & \sum _{u_i\in V(G_1)}{(d_{u_i}^{+}+d_{u_i}^{-})}^2{n}_2\hfill \\ \hfill & \hfill & +\sum _{v_j\in V(G_2)}{(d_{v_j}^{+}+d_{v_j}^{-})}^2{n}_1+2(2m_1)(2m_2)\hfill \\ \hfill & =\hfill & n_2\sum _{u_i\in V(G_1)}(d_{u_i}^{+}{}^2+d_{u_i}^{-}{}^2+2d_{u_i}^{+}{d}_{u_i}^{-})\hfill \\ \hfill & \hfill & +n_1\sum _{v_j\in V(G_2)}(d_{v_j}^{+}{}^2+d_{v_j}^{-}{}^2+2d_{v_j}^{+}{d}_{v_j}^{-})+8m_1{m}_2,\hfill \end{array}$$which gives us the Equation (9)(9) $$M_1(G_1\square G_2)=n_2{M}_1(S_1)+n_1{M}_1(S_2)+8m_1{m}_2$$(9) . □

Theorem 2.3.

Let $S_1(G_1,{\sigma }_1)$ and $S_2(G_2,{\sigma }_2)$ be two signed graphs, with n₁, n₂ and m₁, m₂ being the number of vertices and number of edges of G₁ and G₂, respectively. Then the first Zagreb index of the lexicographic product of G₁ and G₂ in terms of the Zagreb index of their signed graphs is given by: (10) $$M_1(G_1[G_2])=n_2^3{M}_1(S_1)+n_1{M}_1(S_2)+8n_2{m}_1{m}_2$$(10) where $M_1(S_1)=[M_1^{+}(S_1)+M_1^{-}(S_1)+2M_1^{*}(S_1)]$ and $M_1(S_2)=[M_1^{+}(S_2)+M_1^{-}(S_2)+2M_1^{*}(S_2)]$.

Proof.

By applying the results in the Tables 1 and 2 we obtain, $$\begin{array}{ccc}{M}_1(G_1[G_2])\hfill & =\hfill & \sum _{(u_i,v_j)\in V(G_1[G_2])}{\left[d_{(u_i,v_j)}\right]}^2\hfill \\ \hfill & =\hfill & \sum _{(u_i,v_j)\in V(G_1[G_2])}{(n_2{d}_{u_i}+d_{v_j})}^2\hfill \\ \hfill & =\hfill & \sum _{(u_i,v_j)\in V(G_1[G_2])}(n_2^2{d}_{u_i}^2+d_{v_j}^2+2n_2{d}_{u_i}{d}_{v_j})\hfill \\ \hfill & =\hfill & \sum _{u_i\in V(G_1)}{n}_2^2{d}_{u_i}^2\sum _{v_j\in V(G_2)}1+\sum _{v_j\in V(G_2)}{d}_{v_j}^2\sum _{u_i\in V(G_1)}1\hfill \\ \hfill & \hfill & +2n_2\sum _{u_i\in V(G_1)}{d}_{u_i}\sum _{v_j\in V(G_2)}{d}_{v_j}\hfill \\ \hfill & =\hfill & n_2^3\sum _{u_i\in V(G_1)}{d}_{u_i}^2+n_1\sum _{v_j\in V(G_2)}{d}_{v_j}^2+8n_2{m}_1{m}_2\hfill \\ \hfill & =\hfill & n_2^3\sum _{u_i\in V(G_1)}{(d_{u_i}^{+}+d_{u_i}^{-})}^2\hfill \\ \hfill & \hfill & +n_1\sum _{v_j\in V(G_2)}{(d_{v_j}^{+}+d_{v_j}^{-})}^2+8n_2{m}_1{m}_2\hfill \\ \hfill & =\hfill & n_2^3\sum _{u_i\in V(G_1)}(d_{u_i}^{+}{}^2+d_{u_i}^{-}{}^2+2d_{u_i}^{+}{d}_{u_i}^{-})\hfill \\ \hfill & \hfill & +n_1\sum _{v_j\in V(G_2)}(d_{v_j}^{+}{}^2+d_{v_j}^{-}{}^2+2d_{v_j}^{+}{d}_{v_j}^{-})+8n_2{m}_1{m}_2,\hfill \end{array}$$which gives us Equation (10)(10) $$M_1(G_1[G_2])=n_2^3{M}_1(S_1)+n_1{M}_1(S_2)+8n_2{m}_1{m}_2$$(10) . □

Theorem 2.4.

Let $S_1(G_1,{\sigma }_1)$ and $S_2(G_2,{\sigma }_2)$ be two signed graphs, with n₁, n₂ and m₁, m₂ being the number of vertices and number of edges of G₁ and G₂, respectively. Then the first Zagreb index of the strong product of G₁ and G₂ in terms of the Zagreb index of their signed graphs is given by: (11) $$\begin{array}{c}{M}_1(G_1*G_2)=(n_2+4m_2)M_1(S_1)+M_1(S_1)M_1(S_2)\\ +(n_1+4m_1)M_1(S_2)+8m_1{m}_2,\end{array}$$(11) where $M_1(S_1)=[M_1^{+}(S_1)+M_1^{-}(S_1)+2M_1^{*}(S_1)]$ and $M_1(S_2)=[M_1^{+}(S_2)+M_1^{-}(S_2)+2M_1^{*}(S_2)]$.

Proof.

By applying the results in the Tables 1 and 2 we obtain, we have, $$\begin{array}{c}{M}_1(G_1*G_2)=\sum _{(u_i,v_j)\in V(G_1*G_2)}{\left[d_{(u_i,v_j)}\right]}^2\\ =\sum _{(u_i,v_j)\in V(G_1*G_2)}{(d_{u_i}+d_{v_j}+d_{u_i}{d}_{v_j})}^2\\ =\sum _{(u_i,v_j)\in V(G_1*G_2)}(d_{u_i}^2+d_{v_j}^2+d_{u_i}^2{d}_{v_j}^2+2d_{u_i}{d}_{v_j}\\ +2d_{u_i}{d}_{v_j}^2+2d_{u_i}^2{d}_{v_j})\\ =\sum _{u_i\in V(G_1)}{d}_{u_i}^2\sum _{v_j\in V(G_2)}1+\sum _{v_j\in V(G_2)}{d}_{v_j}^2\sum _{u_i\in V(G_1)}1\\ +\sum _{u_i\in V(G_1)}{d}_{u_i}^2\sum _{v_j\in V(G_2)}{d}_{v_j}^2+2\sum _{u_i\in V(G_1)}{d}_{u_i}\sum _{v_j\in V(G_2)}{d}_{v_j}\\ +2\sum _{u_i\in V(G_1)}{d}_{u_i}\sum _{v_j\in V(G_2)}{d}_{v_j}^2+2\sum _{u_i\in V(G_1)}{d}_{u_i}^2\sum _{v_j\in V(G_2)}{d}_{v_j}\\ =\sum _{u_i\in V(G_1)}{d}_{u_i}^2(n_2+2\sum _{v_j\in V(G_2)}{d}_{v_j})+\sum _{u_i\in V(G_1)}{d}_{u_i}^2\sum _{v_j\in V(G_2)}{d}_{v_j}^2\\ +\sum _{v_j\in V(G_2)}{d}_{v_j}^2(n_1+2\sum _{u_i\in V(G_1)}{d}_{u_i})\\ +2\sum _{u_i\in V(G_1)}{d}_{u_i}\sum _{v_j\in V(G_2)}{d}_{v_j}\\ =\sum _{u_i\in V(G_1)}{(d_{u_i}^{+}+d_{u_i}^{-})}^2(n_2+4m_2)+\sum _{u_i\in V(G_1)}{d}_{u_i}^2\sum _{v_j\in V(G_2)}{d}_{v_j}^2\\ +\sum _{v_j\in V(G_2)}{(d_{v_j}^{+}+d_{v_j}^{-})}^2(n_1+4m_1)+8m_1{m}_2\\ =\sum _{u_i\in V(G_1)}(d_{u_i}^{+}{}^2+d_{u_i}^{-}{}^2+2d_{u_i}^{+}{d}_{u_i}^{-})(n_2+4m_2)\\ +\sum _{u_i\in V(G_1)}{(d_{u_i}^{+}+d_{u_i}^{-})}^2\sum _{v_j\in V(G_2)}{(d_{v_j}^{+}+d_{v_j}^{-})}^2\\ +\sum _{v_j\in V(G_2)}(d_{v_j}^{+}{}^2+d_{v_j}^{-}{}^2+2d_{v_j}^{+}{d}_{v_j}^{-})(n_1+4m_1)+8m_1{m}_2,\end{array}$$which gives Equation (11)(11) $$\begin{array}{c}{M}_1(G_1*G_2)=(n_2+4m_2)M_1(S_1)+M_1(S_1)M_1(S_2)\\ +(n_1+4m_1)M_1(S_2)+8m_1{m}_2,\end{array}$$(11) . □

Theorem 2.5.

Let $S_1(G_1,{\sigma }_1)$ and $S_2(G_2,{\sigma }_2)$ be two signed graphs, with n₁, n₂ and m₁, m₂ being the number of vertices and number of edges of G₁ and G₂, respectively. Then the first Zagreb index of the symmetric difference of G₁ and G₂ in terms of the Zagreb index of their signed graphs is given by: (12) $$\begin{array}{c}{M}_1(G_1\oplus G_2)=[n_2^3-8n_2{m}_2]M_1(S_1)+M_1(S_2)[n_1^3-8n_1{m}_1]\\ \hspace{1em}+4M_1(S_1)M_1(S_2)+8m_1{m}_2{n}_1{n}_2,\end{array}$$(12) where $M_1(S_1)=[M_1^{+}(S_1)+M_1^{-}(S_1)+2M_1^{*}(S_1)]$ and $M_1(S_2)=[M_1^{+}(S_2)+M_1^{-}(S_2)+2M_1^{*}(S_2)]$.

Proof.

By applying the results in the Tables 1 and 2 we obtain, $$\begin{array}{c}{M}_1(G_1\oplus G_2)=\sum _{(u_i,v_j)\in V(G_1\oplus G_2)}{\left[d_{(u_i,v_j)}\right]}^2\\ =\sum _{(u_i,v_j)\in V(G_1\oplus G_2)}{(n_2{d}_{u_i}+n_1{d}_{v_j}-2d_{u_i}{d}_{v_j})}^2\\ =\sum _{(u_i,v_j)\in V(G_1\oplus G_2)}(n_2^2{d}_{u_i}^2+n_1^2{d}_{v_j}^2+4d_{u_i}^2{d}_{v_j}^2+2n_1{n}_2{d}_{u_i}{d}_{v_j}\\ -4n_1{d}_{u_i}{d}_{v_j}^2-4n_2{d}_{u_i}^2{d}_{v_j})\\ =n_2^2\sum _{u_i\in V(G_1)}{d}_{u_i}^2\sum _{v_j\in V(G_2)}1+n_1^2\sum _{v_j\in V(G_2)}{d}_{v_j}^2\sum _{u_i\in V(G_1)}1\\ +4\sum _{u_i\in V(G_1)}{d}_{u_i}^2\sum _{v_j\in V(G_2)}{d}_{v_j}^2+2n_1{n}_2\sum _{u_i\in V(G_1)}{d}_{u_i}\sum _{v_j\in V(G_2)}{d}_{v_j}\\ -4n_1\sum _{u_i\in V(G_1)}{d}_{u_i}\sum _{v_j\in V(G_2)}{d}_{v_j}^2-4n_2\sum _{u_i\in V(G_1)}{d}_{u_i}^2\sum _{v_j\in V(G_2)}{d}_{v_j}\\ =\mathrm{}\sum _{u_i\in V(G_1)}{d}_{u_i}^2[n_2^3-4n_2(2m_2)]+\mathrm{}\sum _{v_j\in V(G_2)}{d}_{v_j}^2[n_1^3-4n_1(2m_1)]\\ +4\sum _{u_i\in V(G_1)}{d}_{u_i}^2\sum _{v_j\in V(G_2)}{d}_{v_j}^2+8m_1{m}_2{n}_1{n}_2\\ =\sum _{u_i\in V(G_1)}{(d_{u_i}^{+}+d_{u_i}^{-})}^2[n_2^3-8n_2{m}_2]\\ +\sum _{v_j\in V(G_2)}{(d_{v_j}^{+}+d_{v_j}^{-})}^2[n_1^3-8n_1{m}_1]\\ +4\mathrm{}\sum _{u_i\in V(G_1)}{(d_{u_i}^{+}+d_{u_i}^{-})}^2\mathrm{}\sum _{v_j\in V(G_2)}{(d_{v_j}^{+}+d_{v_j}^{-})}^2+8m_1{m}_2{n}_1{n}_2,\end{array}$$which gives us Equation (12)(12) $$\begin{array}{c}{M}_1(G_1\oplus G_2)=[n_2^3-8n_2{m}_2]M_1(S_1)+M_1(S_2)[n_1^3-8n_1{m}_1]\\ \hspace{1em}+4M_1(S_1)M_1(S_2)+8m_1{m}_2{n}_1{n}_2,\end{array}$$(12) . □

Theorem 2.6.

Let $(S_1,{\sigma }_1)$ and $(S_2,{\sigma }_2)$ be two signed graphs and their underlying graphs be G₁ and G₂, respectively. Then the second Zagreb index of the tensor product of G₁ and G₂, in terms of the Zagreb index of their signed graphs is given by: (13) $$M_2(G_1\otimes G_2)=2M_2(S_1)M_2(S_2)$$(13) where $M_2(S_1)=[M_2^{+}(S_1)+M_2^{-}(S_1)+2M_2^{*}(S_1)]$ and $M_2(S_2)=[M_2^{+}(S_2)+M_2^{-}(S_2)+2M_2^{*}(S_2)]$.

Proof.

By applying the results in the Tables 1, 2, Lemma 1.2 and Equation (3)(3) $$|E(G_1\otimes G_2)|=2|E(G_1)||E(G_2)|.$$(3) we have: $$\begin{array}{ccc}{M}_2(G_1\otimes G_2)\hfill & =\hfill & \sum _{(u_i,a_i)(v_j,b_j)\in E(G_1\otimes G_2)}\left[d_{(u_i,a_i)}{d}_{(v_j,b_j)}\right]\hfill \\ \hfill & =\hfill & \sum _{(u_i,a_i)(v_j,b_j)\in E(G_1\otimes G_2)}(d_{u_i},d_{a_i})(d_{v_j},d_{b_j})\hfill \\ \hfill & =\hfill & \sum _{(u_i,a_i)(v_j,b_j)\in E(G_1\otimes G_2)}{d}_{u_i}{d}_{v_j}{d}_{a_i}{d}_{b_j}\hfill \\ \hfill & =\hfill & 2\sum _{u_i{v}_j\in E(G_1)}{d}_{u_i}{d}_{v_j}\sum _{a_i{b}_j\in E(G_2)}{d}_{a_i}{d}_{b_j}\hfill \\ \hfill & =\hfill & 2\sum _{u_i{v}_j\in E(G_1)}(d_{u_i}^{+}+d_{u_i}^{-})(d_{v_j}^{+}+d_{v_j}^{-})\hfill \\ \hfill & \hfill & \times \sum _{a_i{b}_j\in E(G_2)}(d_{a_i}^{+}+d_{a_i}^{-})(d_{b_j}^{+}+d_{b_j}^{-})\hfill \\ \hfill & =\hfill & 2\sum _{u_i{v}_j\in E(G_1)}(d_{u_i}^{+}{d}_{v_j}^{+})+(d_{u_i}^{+}{d}_{v_j}^{-})\hfill \\ \hfill & \hfill & +(d_{u_i}^{-}{d}_{v_j}^{+})+(d_{u_i}^{-}{d}_{v_j}^{-})\mathrm{}\mathrm{}\sum _{a_i{b}_j\in E(G_2)}(d_{a_i}^{+}{d}_{b_j}^{+})+(d_{a_i}^{+}{d}_{b_j}^{-})\hfill \\ \hfill & \hfill & +(d_{a_i}^{-}{d}_{b_j}^{+})+(d_{a_i}^{-}{d}_{b_j}^{-}),\hfill \end{array}$$which gives us Equation (13)(13) $$M_2(G_1\otimes G_2)=2M_2(S_1)M_2(S_2)$$(13) . □

Theorem 2.7.

Let $S_1(G_1,{\sigma }_1)$ and $S_2(G_2,{\sigma }_2)$ be two signed graphs, with n₁, n₂ and m₁, m₂ being the number of vertices and number of edges of G₁ and G₂, respectively. Then the second Zagreb index of the Cartesian product of G₁ and G₂, in terms of the Zagreb index of their signed graphs is given by: (14) $$\begin{array}{c}{M}_2(G_1\square G_2)=m_2{M}_1(S_1)+2m_1{\mathrm{M}}_1({\mathrm{S}}_2)+n_1{M}_2(S_2)\\ +m_1{M}_1(S_2)+2m_2{M}_1(S_1)+n_2{M}_2(S_1),\end{array}$$(14) where $M_1(S_1)=[M_1^{+}(S_1)+M_1^{-}(S_1)+2M_1^{*}(S_1)],M_1(S_2)=[M_1^{+}(S_2)+M_1^{-}(S_2)+2M_1^{*}(S_2)],M_2(S_1)=[M_2^{+}(S_1)+M_2^{-}(S_1)+2M_2^{*}(S_1)],M_2(S_2)=[M_2^{+}(S_2)+M_2^{-}(S_2)+2M_2^{*}(S_2)]$ and ${\mathrm{M}}_1({\mathrm{S}}_2)=[M_1^{+}(S_2)+M_1^{-}(S_2)+2M_1^{*}(S_2)]$.

Proof.

By applying the results in the Table 1, Lemma 1.2 and Equation (4)(4) $$|E(G_1\square G_2)|=|E(G_1)||V(G_2)|+|V(G_1)||E(G_2)|.$$(4) we have: $$\begin{array}{c}{M}_2(G_1\square G_2)=\sum _{(u_i,a_i)(v_j,b_j)\in E(G_1\square G_2)}\left[d_{(u_i,a_i)}{d}_{(v_j,b_j)}\right]\\ =\sum _{x_i\in V(G_1)}\sum _{(a_i{b}_j)\in E(G_2)}{d}_{G_1\square G_2}(x_i,a_i)d_{G_1\square G_2}(x_i,b_j)\\ \mathrm{}+\mathrm{}\sum _{y_j\in V(G_2)}\sum _{(u_i{v}_j)\in E(G_1)}\mathrm{}{d}_{G_1\square G_2}(u_i,y_j)d_{G_1\square G_2}(v_j,y_j)\\ =\sum _{x_i\in V(G_1)}\sum _{(a_i{b}_j)\in E(G_2)}(d_{x_i}+d_{a_i})(d_{x_i}+d_{b_j})\\ +\sum _{y_j\in V(G_2)}\sum _{(u_i{v}_j)\in E(G_1)}(d_{u_i}+d_{y_j})(d_{v_j}+d_{y_j})\\ =\sum _{x_i\in V(G_1)}\sum _{(a_i{b}_j)\in E(G_2)}{(d_{x_i})}^2+(d_{x_i}{d}_{b_j})+(d_{a_i}{d}_{x_i})+(d_{a_i}{d}_{b_j})\\ +\sum _{y_j\in V(G_2)}\sum _{(u_i{v}_j)\in E(G_1)}{(d_{y_j})}^2+(d_{y_j}{d}_{v_j})+(d_{y_j}{d}_{u_i})+(d_{u_i}{d}_{v_j})\\ =\mathrm{}\sum _{x_i\in V(G_1)}{(d_{x_i})}^2\mathrm{}\sum _{(a_i{b}_j)\in E(G_2)}1+\mathrm{}\sum _{x_i\in V(G_1)}{d}_{x_i}\mathrm{}\sum _{(a_i{b}_j)\in E(G_2)}(d_{a_i}+d_{b_j})\\ +\sum _{x_i\in V(G_1)}\sum _{(a_i{b}_j)\in E(G_2)}(d_{a_i}{d}_{b_j})+\sum _{y_j\in V(G_2)}{(d_{y_j})}^2\sum _{(u_i{v}_j)\in E(G_1)}1\\ +\sum _{y_j\in V(G_2)}{d}_{y_j}\sum _{(u_i{v}_j)\in E(G_1)}(d_{v_j}+d_{u_i})\\ +\sum _{y_j\in V(G_2)}\sum _{(u_i{v}_j)\in E(G_1)}(d_{u_i}{d}_{v_j})\\ =\sum _{x_i\in V(G_1)}{(d_{x_i}^{+}+d_{x_i}^{-})}^2{m}_2\\ +2m_1\sum _{(a_i{b}_j)\in E(G_2)}(d_{a_i}^{+}+d_{a_i}^{-})+(d_{b_j}^{+}+d_{b_j}^{-})\\ +n_1\sum _{(a_i{b}_j)\in E(G_2)}(d_{a_i}^{+}+d_{a_i}^{-})(d_{b_j}^{+}+d_{b_j}^{-})\\ +\sum _{y_j\in V(G_2)}{(d_{y_j}^{+}+d_{y_j}^{-})}^2{m}_1\\ +2m_2\sum _{(u_i{v}_j)\in E(G_1)}{d}_{v_j}^{+}+d_{v_j}^{-}+d_{u_i}^{+}+d_{u_i}^{-}\\ +n_2\sum _{(u_i{v}_j)\in E(G_1)}(d_{u_i}^{+}+d_{u_i}^{-})(d_{v_j}^{+}+d_{v_j}^{-})\\ =m_2\sum _{x_i\in V(G_1)}(d_{x_i}^{+}{}^2+2d_{x_i}^{+}{d}_{x_i}^{-}+d_{x_i}^{-}{}^2)\\ +2m_1\sum _{(a_i{b}_j)\in E(G_2)}(d_{a_i}^{+}+d_{b_j}^{+}+d_{a_i}^{-}+d_{b_j}^{-})\\ +n_1\sum _{(a_i{b}_j)\in E(G_2)}(d_{a_i}^{+}{d}_{b_j}^{+}+d_{a_i}^{+}{d}_{b_j}^{-}+d_{a_i}^{-}{d}_{b_j}^{+}+d_{a_i}^{-}{d}_{b_j}^{-})\\ +m_1\sum _{y_j\in V(G_2)}(d_{y_j}^{+}{}^2+2d_{y_j}^{+}{d}_{y_j}^{-}+d_{y_j}^{-}{}^2)\\ +2m_2\sum _{(u_i{v}_j)\in E(G_1)}(d_{u_i}^{+}+d_{v_j}^{+}+d_{u_i}^{-}+d_{v_j}^{-})\\ +n_2\sum _{(u_i{v}_j)\in E(G_1)}(d_{u_i}^{+}{d}_{v_j}^{+}+d_{u_i}^{+}{d}_{v_j}^{-}+d_{u_i}^{-}{d}_{v_j}^{+}+d_{u_i}^{-}{d}_{v_j}^{-}),\end{array}$$which gives Equation (14)(14) $$\begin{array}{c}{M}_2(G_1\square G_2)=m_2{M}_1(S_1)+2m_1{\mathrm{M}}_1({\mathrm{S}}_2)+n_1{M}_2(S_2)\\ +m_1{M}_1(S_2)+2m_2{M}_1(S_1)+n_2{M}_2(S_1),\end{array}$$(14) . □

Theorem 2.8.

Let $S_1(G_1,{\sigma }_1)$ and $S_2(G_2,{\sigma }_2)$ be two signed graphs, with n₁, n₂ and m₁, m₂ being the number of vertices and number of edges of G₁ and G₂, respectively. Then the second Zagreb index of the lexicographic product of G₁ and G₂, in terms of the Zagreb index of their signed graphs is given by: (15) $$\begin{array}{c}{M}_2(G_1[G_2])=m_2{n}_2^2{M}_1(S_1)+2n_2{m}_1{\mathrm{M}}_1({\mathrm{S}}_2)+n_1{M}_2(S_2)\\ +n_2^4{M}_2(S_1)+2m_2{n}_2^2{\mathrm{M}}_1({\mathrm{S}}_1)+4m_2^2{m}_1,\end{array}$$(15) where $M_1(S_1)=[M_1^{+}(S_1)+M_1^{-}(S_1)+2M_1^{*}(S_1)],M_1(S_2)=[M_1^{+}(S_2)+M_1^{-}(S_2)+2M_1^{*}(S_2)],M_2(S_1)=[M_2^{+}(S_1)+M_2^{-}(S_1)+2M_2^{*}(S_1)],M_2(S_2)=[M_2^{+}(S_2)+M_2^{-}(S_2)+2M_2^{*}(S_2)]$ and ${\mathrm{M}}_1({\mathrm{S}}_2)=[M_1^{+}(S_2)+M_1^{-}(S_2)]$.

Proof.

By applying the results in the Table 1 and Lemma 1.2, Equation (5)(5) $$|E(G_1\left[G_2\right])|=|E(G_1)||V(G_2){|}^2+|E(G_2)\left|\right|V(G_1)|.$$(5) we have: $$\begin{array}{c}{M}_2(G_1[G_2])=\sum _{(u_i,a_i)(v_j,b_j)\in E(G_1[G_2])}\left[d_{(u_i,a_i)}{d}_{(v_j,b_j)}\right]\\ =\sum _{x_i\in V(G_1)}\sum _{(a_i{b}_j)\in E(G_2)}{d}_{G_1\left[G_2\right]}(x_i,a_i)d_{G_1\left[G_2\right]}(x_i,b_j)\\ +\sum _{y_j\in V(G_2)}\sum _{z_k\in V(G_2)}\sum _{(u_i{v}_j)\in E(G_1)}{d}_{G_1\left[G_2\right]}(u_i,y_j)d_{G_1\left[G_2\right]}(v_j,z_k)\\ =\sum _{x_i\in V(G_1)}\sum _{(a_i{b}_j)\in E(G_2)}(n_2{d}_{x_i}+d_{a_i})(n_2{d}_{x_i}+d_{b_j})\\ +\sum _{y_j\in V(G_2)}\sum _{z_k\in V(G_2)}\sum _{(u_i{v}_j)\in E(G_1)}(n_2{d}_{u_i}+d_{y_j})(n_2{d}_{v_j}+d_{z_k})\\ =\sum _{x_i\in V(G_1)}\sum _{(a_i{b}_j)\in E(G_2)}[n_2^2{d}_{x_i}^2+n_2{d}_{x_i}{d}_{b_j}+n_2{d}_{x_i}{d}_{a_i}+d_{a_i}{d}_{b_j}]\\ +\sum _{y_j\in V(G_2)}\sum _{z_k\in V(G_2)}\sum _{(u_i{v}_j)\in E(G_1)}[n_2^2{d}_{u_i}{d}_{v_j}+n_2{d}_{u_i}{d}_{z_k}\\ +n_2{d}_{y_j}{d}_{v_j}+d_{y_j}{d}_{z_k}]\\ =n_2^2\sum _{x_i\in V(G_1)}{d}_{x_i}^2\sum _{(a_i{b}_j)\in E(G_2)}1\\ +n_2\sum _{x_i\in V(G_1)}{d}_{x_i}\sum _{(a_i{b}_j)\in E(G_2)}(d_{a_i}+d_{b_j})\end{array}$$ $$\begin{array}{c}+n_1\sum _{(a_i{b}_j)\in E(G_2)}{d}_{a_i}{d}_{b_j}\\ +n_2^2\sum _{y_j\in V(G_2)}1\sum _{z_k\in V(G_2)}1\sum _{(u_i{v}_j)\in E(G_1)}{d}_{u_i}{d}_{v_j}\\ +n_2\sum _{y_j\in V(G_2)}1\sum _{z_k\in V(G_2)}{d}_{z_k}\sum _{(u_i{v}_j)\in E(G_1)}{d}_{u_i}+d_{v_j}\\ +\sum _{y_j\in V(G_2)}{d}_{y_j}\sum _{z_k\in V(G_2)}{d}_{z_k}\sum _{(u_i{v}_j)\in E(G_1)}1\\ =n_2^2\sum _{x_i\in V(G_1)}{(d_{x_i}^{+}+d_{x_i}^{-})}^2{m}_2\\ +n_{2}2m_1\sum _{(a_i{b}_j)\in E(G_2)}(d_{a_i}^{+}+d_{a_i}^{-})+(d_{b_j}^{+}+d_{b_j}^{-})\\ +n_1\sum _{(a_i{b}_j)\in E(G_2)}(d_{a_i}^{+}+d_{a_i}^{-})(d_{b_j}^{+}{d}_{b_j}^{-})\\ +n_2^4\sum _{(u_i{v}_j)\in E(G_1)}(d_{u_i}^{+}+d_{u_i}^{-})(d_{v_j}^{+}+d_{v_j}^{-})\\ +n_2^{2}2m_2\sum _{(u_i{v}_j)\in E(G_1)}(d_{u_i}^{+}+d_{u_i}^{-})+(d_{v_j}^{+}+d_{v_j}^{-})+4m_2^2{m}_1\\ =n_2^2\sum _{x_i\in V(G_1)}(d_{x_i}^{+}{}^2+2d_{x_i}^{+}{d}_{x_i}^{-}+d_{x_i}^{-}{}^2)m_2\\ +n_{2}2m_1\sum _{(a_i{b}_j)\in E(G_2)}(d_{a_i}^{+}+d_{b_j}^{+}+d_{a_i}^{-}+d_{b_j}^{-})\\ +n_1\sum _{(a_i{b}_j)\in E(G_2)}(d_{a_i}^{+}{d}_{b_j}^{+}+d_{a_i}^{+}{d}_{b_j}^{-}+d_{a_i}^{-}{d}_{b_j}^{+}+d_{a_i}^{-}{d}_{b_j}^{-})\\ +n_2^4\sum _{(u_i{v}_j)\in E(G_1)}(d_{u_i}^{+}{d}_{v_j}^{+}+d_{u_i}^{+}{d}_{v_j}^{-}+d_{u_i}^{-}{d}_{v_j}^{+}+d_{u_i}^{-}{d}_{v_j}^{-})\\ +2m_2{n}_2^2\sum _{(u_i{v}_j)\in E(G_1)}(d_{u_i}^{+}+d_{v_j}^{+}+d_{u_i}^{-}+d_{v_j}^{-})+4m_2^2{m}_1,\end{array}$$which gives Equation (15)(15) $$\begin{array}{c}{M}_2(G_1[G_2])=m_2{n}_2^2{M}_1(S_1)+2n_2{m}_1{\mathrm{M}}_1({\mathrm{S}}_2)+n_1{M}_2(S_2)\\ +n_2^4{M}_2(S_1)+2m_2{n}_2^2{\mathrm{M}}_1({\mathrm{S}}_1)+4m_2^2{m}_1,\end{array}$$(15) . □

Theorem 2.9.

Let $S_1(G_1,{\sigma }_1)$ and $S_2(G_2,{\sigma }_2)$ be two signed graphs, with n₁, n₂ and m₁, m₂ being the number of vertices and number of edges of G₁ and G₂, respectively. Then the second Zagreb index of the strong product of G₁ and G₂, in terms of the Zagreb index of their signed graphs is given by: (16) $$\begin{array}{c}{M}_2(G_1*G_2)=3m_2{M}_1(S_1)+3m_1{M}_1(S_2)+(n_2+6m_2)M_2(S_1)\\ +(2m_1+n_1)M_2(S_2)+3M_1(S_1)M_1(S_2)\\ +3M_1(S_1)M_2(S_2)+3M_1(S_2)M_2(S_1)\\ +2M_2(S_1)M_2(S_2).\end{array}$$(16) where $M_1(S_1)=[M_1^{+}(S_1)+M_1^{-}(S_1)+2M_1^{*}(S_1)],M_1(S_2)=[M_1^{+}(S_2)+M_1^{-}(S_2)+2M_1^{*}(S_2)],M_2(S_1)=[M_2^{+}(S_1)+M_2^{-}(S_1)+2M_2^{*}(S_1)]$ and $M_2(S_2)=[M_2^{+}(S_2)+M_2^{-}(S_2)+2M_2^{*}(S_2)]$.

Proof.

By applying the results in the Tables 1, 2 and Lemma 1.2, Equation (6)(6) $$\begin{array}{c}|E(G_1*G_2)|=|E(G_1)||V(G_2)|+|E(G_2)||V(G_1)|\\ +2|E(G_1)||E(G_2)|.\end{array}$$(6) we have: $$\begin{array}{ccc}{M}_2(G_1*G_2)\hfill & =\hfill & \sum _{(u_i,a_i)(v_j,b_j)\in E(G_1*G_2)}\left[d_{(u_i,a_i)}{d}_{(v_j,b_j)}\right]\hfill \\ \hfill & =\hfill & \sum _{x_i\in V(G_1)}\sum _{(a_i{b}_j)\in E(G_2)}{d}_{G_1*G_2}(x_i,a_i)d_{G_1*G_2}(x_i,b_j)\hfill \\ \hfill & \hfill & +\sum _{y_j\in V(G_2)}\sum _{(u_i{v}_j)\in E(G_1)}{d}_{G_1*G_2}(u_i,y_j)d_{G_1*G_2}(v_j,y_j)\hfill \\ \hfill & \hfill & +2\sum _{(u_i{v}_j)\in E(G_1)}\sum _{(a_i{b}_j)\in E(G_2)}{d}_{G_1*G_2}(u_i,a_i)d_{G_1*G_2}(v_j,b_j)\hfill \\ \hfill & =\hfill & \sum _{x_i\in V(G_1)}\sum _{(a_i{b}_j)\in E(G_2)}(d_{x_i}+d_{a_i}+d_{x_i}{d}_{a_i})(d_{x_i}+d_{b_j}+d_{x_i}{d}_{b_j})\hfill \\ \hfill & \hfill & +\mathrm{}\sum _{y_j\in V(G_2)}\sum _{(u_i{v}_j)\in E(G_1)}\mathrm{}(d_{u_i}\mathrm{}+\mathrm{}{d}_{y_j}\mathrm{}+\mathrm{}{d}_{u_i}{d}_{y_j})(d_{v_j}\mathrm{}+\mathrm{}{d}_{y_j}\mathrm{}+\mathrm{}{d}_{v_j}{d}_{y_j})\hfill \\ \hfill & \hfill & +2\sum _{(u_i{v}_j)\in E(G_1)}\sum _{(a_i{b}_j)\in E(G_2)}(d_{u_i}+d_{a_i}+d_{u_i}{d}_{a_i})\hfill \\ \hfill & \hfill & \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}(d_{v_j}+d_{b_j}+d_{v_j}{d}_{b_j})\hfill \end{array}$$ $$\begin{array}{c}\begin{array}{ccc}\hfill & =\hfill & \sum _{x_i\in V(G_1)}\sum _{(a_i{b}_j)\in E(G_2)}[d_{x_i}^2+d_{x_i}{d}_{b_j}+d_{x_i}^2{d}_{b_j}+d_{a_i}{d}_{x_i}+d_{a_i}{d}_{b_j}\hfill \\ \hfill & \hfill & +d_{a_i}{d}_{x_i}{d}_{b_j}+d_{x_i}^2{d}_{a_i}+d_{x_i}{d}_{a_i}{d}_{b_j}+d_{x_i}^2{d}_{a_i}{d}_{b_j})]\hfill \\ \hfill & \hfill & +\sum _{y_j\in V(G_2)}\sum _{(u_i{v}_j)\in E(G_1)}[d_{u_i}{d}_{v_j}+d_{u_i}{d}_{y_j}+d_{u_i}{d}_{v_j}{d}_{y_j}+d_{v_j}{d}_{y_j}+d_{y_j}^2\hfill \\ \hfill & \hfill & +d_{v_j}{d}_{y_j}^2+d_{u_i}{d}_{v_j}{d}_{y_j}+d_{u_i}{d}_{y_j}^2+d_{u_i}{d}_{v_j}{d}_{y_j}^2]\hfill \end{array}\\ \begin{array}{ccc}\hfill & \hfill & +2\sum _{(u_i{v}_j)\in E(G_1)}\sum {}_{(a_i{b}_j)\in E(G_2)}[d_{u_i}{d}_{v_j}+d_{u_i}{d}_{b_j}+d_{u_i}{d}_{v_j}{d}_{b_j}+d_{a_i}{d}_{v_j}\hfill \\ \hfill & \hfill & \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+d_{a_i}{d}_{b_j}+d_{a_i}{d}_{v_j}{d}_{b_j}+d_{u_i}{d}_{a_i}{d}_{v_j}+d_{u_i}{d}_{a_i}{d}_{b_j}+d_{u_i}{d}_{v_j}{d}_{a_i}{d}_{b_j}]\hfill \\ \hfill & =\hfill & \sum _{x_i\in V(G_1)}\sum _{(a_i{b}_j)\in E(G_2)}[d_{x_i}^2(1+d_{b_j}+d_{a_i}+d_{a_i}{d}_{b_j})\hfill \\ \hfill & \hfill & +d_{x_i}(d_{a_i}+d_{b_j}+2d_{a_i}{d}_{b_j})+d_{a_i}{d}_{b_j}]\hfill \\ \hfill & \hfill & +\sum _{y_j\in V(G_2)}\sum _{(u_i{v}_j)\in E(G_1)}[d_{y_j}^2(1+d_{v_j}+d_{u_i}+d_{u_i}{d}_{v_j})\hfill \\ \hfill & \hfill & +d_{y_j}(d_{u_i}+d_{v_j}+2d_{u_i}{d}_{v_j})+d_{u_i}{d}_{v_j}]\hfill \end{array}\end{array}$$ $$\begin{array}{ccc}\hfill & \hfill & +2\sum _{(u_i{v}_j)\in E(G_1)}\sum _{(a_i{b}_j)\in E(G_2)}[d_{u_i}{d}_{v_j}(1+d_{a_i}{d}_{b_j}+d_{a_i}+d_{b_j})\hfill \\ \hfill & \hfill & +d_{a_i}{d}_{b_j}(1+d_{u_i}+d_{v_j})+d_{u_i}{d}_{b_j}+d_{a_i}{d}_{v_j}]\hfill \\ \hfill & =\hfill & \sum _{x_i\in V(G_1)}{d}_{x_i}^2[\sum _{(a_i{b}_j)\in E(G_2)}1+\sum _{(a_i{b}_j)\in E(G_2)}(d_{a_i}+d_{b_j})\hfill \\ \hfill & \hfill & \hspace{1em}+\sum _{(a_i{b}_j)\in E(G_2)}(d_{a_i}{d}_{b_j})]\hfill \\ \hfill & \hfill & +\sum {}_{x_i\in V(G_1)}{d}_{x_i}[\sum _{(a_i{b}_j)\in E(G_2)}(d_{a_i}+d_{b_j})\hfill \\ \hfill & \hfill & \hspace{1em}+2\sum _{(a_i{b}_j)\in E(G_2)}(d_{a_i}{d}_{b_j})]\hfill \\ \hfill & \hfill & +\sum _{x_i\in V(G_1)}1\sum _{(a_i{b}_j)\in E(G_2)}(d_{a_i}{d}_{b_j})\hfill \\ \hfill & \hfill & +\sum {}_{y_j\in V(G_2)}{d}_{y_j}^2[\sum _{(u_i{v}_j)\in E(G_1)}1+\sum _{(u_i{v}_j)\in E(G_1)}(d_{u_i}+d_{v_j})\hfill \\ \hfill & \hfill & \hspace{1em}+\sum _{(u_i{v}_j)\in E(G_1)}(d_{u_i}{d}_{v_j})]\hfill \end{array}$$ $$\begin{array}{ccc}\hfill & \hfill & +\sum _{y_j\in V(G_2)}{d}_{y_j}[\sum _{(u_i{v}_j)\in E(G_1)}(d_{u_i}+d_{v_j})\hfill \\ \hfill & \hfill & \hspace{1em}+2\sum _{(u_i{v}_j)\in E(G_1)}{d}_{u_i}{d}_{v_j}]\hfill \\ \hfill & \hfill & +\sum _{y_j\in V(G_2)}1\sum _{(u_i{v}_j)\in E(G_1)}(d_{u_i}{d}_{v_j})2\sum _{(u_i{v}_j)\in E(G_1)}(d_{u_i}{d}_{v_j})\hfill \\ \hfill & \hfill & \hspace{1em}\times \sum _{(a_i{b}_j)\in E(G_2)}(1+d_{a_i}+d_{b_j})\hfill \\ \hfill & \hfill & +2\sum _{(u_i{v}_j)\in E(G_1)}(d_{u_i}{d}_{v_j})\sum _{(a_i{b}_j)\in E(G_2)}(d_{a_i}{d}_{b_j})\hfill \\ \hfill & \hfill & +2\sum _{(u_i{v}_j)\in E(G_1)}(1+d_{u_i}+d_{v_j})\sum _{(a_i{b}_j)\in E(G_2)}(d_{a_i}{d}_{b_j})\hfill \\ \hfill & \hfill & +2\times \frac{1}{2}\left[\sum _{(u_i{v}_j)\in E(G_1)}\sum _{(a_i{b}_j)\in E(G_2)}(d_{u_i}+d_{v_j})(d_{a_i}+d_{b_j})\right]\hfill \\ \hfill & =\hfill & \sum _{x_i\in V(G_1)}{(d{x}_i^{+}+d{x}_i^{-})}^2{m}_2\hfill \\ \hfill & \hfill & +2m_1\sum _{(a_i{b}_j)\in E(G_2)}(d{a}_i^{+}+d{a}_i^{-}+d{b}_j^{+}+d{b}_j^{-})\hfill \\ \hfill & \hfill & +\sum _{x_i\in V(G_1)}{(d{x}_i^{+}+d{x}_i^{-})}^2\hfill \end{array}$$ $$\begin{array}{ccc}\hfill & \hfill & \hspace{1em}\times \sum _{(a_i{b}_j)\in E(G_2)}(d{a}_i^{+}+d{a}_i^{-}+d{b}_j^{+}+d{b}_j^{-})\hfill \\ \hfill & \hfill & +n_1\sum _{(a_i{b}_j)\in E(G_2)}(d{a}_i^{+}+d{a}_i^{-})(d{b}_j^{+}+d{b}_j^{-})\hfill \\ \hfill & \hfill & +4m_1\sum _{(a_i{b}_j)\in E(G_2)}(d{a}_i^{+}+d{a}_i^{-})(d{b}_j^{+}+d{b}_j^{-})\hfill \\ \hfill & \hfill & +\sum _{x_i\in V(G_1)}{(d{x}_i^{+}+d{x}_i^{-})}^2\hfill \\ \hfill & \hfill & \hspace{1em}\times \sum _{(a_i{b}_j)\in E(G_2)}(d{a}_i^{+}+d{a}_i^{-})(d{b}_j^{+}+d{b}_j^{-})\hfill \\ \hfill & \hfill & +n_2\sum _{(u_i{v}_j)\in E(G_1)}(d{u}_i^{+}+d{u}_i^{-})(d{v}_j^{+}+d{v}_j^{-})\hfill \\ \hfill & \hfill & \hspace{1em}+2m_2\sum _{(u_i{v}_j)\in E(G_1)}(d{u}_i^{+}+d{u}_i^{-}+d{v}_j^{+}+d{v}_j^{-})\hfill \\ \hfill & \hfill & +4m_2\sum _{(u_i{v}_j)\in E(G_1)}(d{u}_i^{+}+d{u}_i^{-})(d{v}_j^{+}+d{v}_j^{-})\hfill \\ \hfill & \hfill & +\sum _{y_j\in V(G_2)}{(d{y}_j^{+}+d{y}_j^{-})}^2\hfill \\ \hfill & \hfill & \hspace{1em}\times \sum _{(u_i{v}_j)\in E(G_1)}(d{u}_i^{+}+d{u}_i^{-}+d{v}_j^{+}+d{v}_j^{-})\hfill \\ \hfill & \hfill & +\sum _{y_j\in V(G_2)}{(d{y}_j^{+}+d{y}_j^{-})}^2\hfill \end{array}$$ $$\begin{array}{c}\begin{array}{ccc}\hfill & \hfill & \hspace{1em}\times \sum _{(u_i{v}_j)\in E(G_1)}(d{u}_i^{+}+d{u}_i^{-})(d{v}_j^{+}+d{v}_j^{-})\hfill \\ \hfill & \hfill & +m_1\sum _{y_j\in V(G_2)}{(d{y}_j^{+}+d{y}_j^{-})}^2\hfill \\ \hfill & \hfill & \hspace{1em}+2m_2\sum _{(u_i{v}_j)\in E(G_1)}(d{u}_i^{+}+d{u}_i^{-})(d{v}_j^{+}+d{v}_j^{-})\hfill \\ \hfill & \hfill & +2\sum _{(u_i{v}_j)\in E(G_1)}(d{u}_i^{+}+d{u}_i^{-})(d{v}_j^{+}+d{v}_j^{-})\hfill \\ \hfill & \hfill & \hspace{1em}\times \sum _{(a_i{b}_j)\in E(G_2)}(d{a}_i^{+}+d{a}_i^{-}+d{b}_j^{+}+d{b}_j^{-})\hfill \\ \hfill & \hfill & +2m_1\sum _{(a_i{b}_j)\in E(G_2)}(d{a}_i^{+}+d{a}_i^{-})(d{b}_j^{+}+d{b}_j^{-})\hfill \\ \hfill & \hfill & +2\sum _{(u_i{v}_j)\in E(G_1)}(d{u}_i^{+}+d{u}_i^{-}+d{v}_j^{+}+d{v}_j^{-})\hfill \end{array}\\ \begin{array}{ccc}\hfill & \hfill & \hspace{1em}\times \sum _{(a_i{b}_j)\in E(G_2)}(d{a}_i^{+}+d{a}_i^{-})(d{b}_j^{+}+d{b}_j^{-})\hfill \\ \hfill & \hfill & +2\sum _{(u_i{v}_j)\in E(G_1)}(d{u}_i^{+}+d{u}_i^{-})(d{v}_j^{+}+d{v}_j^{-})\hfill \\ \hfill & \hfill & \hspace{1em}\times \sum _{(a_i{b}_j)\in E(G_2)}(d{a}_i^{+}+d{a}_i^{-})(d{b}_j^{+}+d{b}_j^{-})\hfill \\ \hfill & \hfill & +\sum _{(u_i{v}_j)\in E(G_1)}(d{u}_i^{+}+d{u}_i^{-})+(d{v}_j^{+}+d{v}_j^{-})\hfill \\ \hfill & \hfill & \hspace{1em}\times \sum _{(a_i{b}_j)\in E(G_2)}(d{a}_i^{+}+d{a}_i^{-})+(d{b}_j^{+}+d{b}_j^{-})\hfill \end{array}\end{array}$$which gives Equation (16)(16) $$\begin{array}{c}{M}_2(G_1*G_2)=3m_2{M}_1(S_1)+3m_1{M}_1(S_2)+(n_2+6m_2)M_2(S_1)\\ +(2m_1+n_1)M_2(S_2)+3M_1(S_1)M_1(S_2)\\ +3M_1(S_1)M_2(S_2)+3M_1(S_2)M_2(S_1)\\ +2M_2(S_1)M_2(S_2).\end{array}$$(16) . □

Theorem 2.10.

Let $S_1(G_1,{\sigma }_1)$ and $S_2(G_2,{\sigma }_2)$ be two signed graphs, with n₁, n₂ and m₁, m₂ being the number of vertices and number of edges of G₁ and G₂ respectively. Then the second Zagreb index of the symmetric difference of G₁ and G₂, in terms of the Zagreb index of their signed graphs is given by: (17) $$\begin{array}{c}{M}_2(G_1\oplus G_2)=4n_2^2{m}_1^2{m}_2+4n_1^2{m}_2^2{m}_1\\ +(2m_1{n}_1^2{n}_2-8m_1^2{n}_2)M_1(S_2)\\ +(n_1^4-8n_1^2{m}_1+16m_1^2)M_2(S_2)\\ +(n_2^4-8n_2^2{m}_2+16m_2^2)M_2(S_1)\\ +(2m_2{n}_1{n}_2^2+8n_1{m}_2^2)M_1(S_1)\\ +2n_1{n}_2{M}_1(S_1)M_1(S_2)\\ -8n_2{M}_2(S_1)M_1(S_2)-8n_1{M}_1(S_1)M_2(S_2)\\ -16M_2(S_1)M_2(S_2),\end{array}$$(17) where $M_1(S_1)=[M_1^{+}(S_1)+M_1^{-}(S_1)+2M_1^{*}(S_1)],M_1(S_2)=[M_1^{+}(S_2)+M_1^{-}(S_2)+2M_1^{*}(S_2)],M_2(S_1)=[M_2^{+}(S_1)+M_2^{-}(S_1)+2M_2^{*}(S_1)]$ and $M_2(S_2)=[M_2^{+}(S_2)+M_2^{-}(S_2)+2M_2^{*}(S_2)]$.

Proof.

By applying the results in the Tables 1, 2 and Lemma 1.2, Equation (7)(7) $$\begin{array}{c}|E(G_1\oplus G_2)|=|E(G_1)||V(G_2){|}^2+|E(G_2)\left|\right|V(G_1){|}^2\\ -4|E(G_1)||E(G_2)|.\end{array}$$(7) we have: $$\begin{array}{c}\begin{array}{ccc}{M}_2(G_1\oplus G_2)\hfill & =\hfill & \sum _{(u_i,a_i)(v_j,b_j)\in E(G_1\oplus G_2)}\left[d_{(u_i,a_i)}{d}_{(v_j,b_j)}\right]\hfill \\ \hfill & =\hfill & \sum _{x_i\in V(G_1)}\sum _{y_j\in V(G_1)}\sum _{(a_i{b}_j)\in E(G_2)}{d}_{G_1\oplus G_2}(x_i,a_i)d_{G_1\oplus G_2}(y_j,b_j)\hfill \\ \hfill & \hfill & +\mathrm{}\sum _{z_k\in V(G_2)}\mathrm{}\sum _{w_l\in V(G_2)}\mathrm{}\sum _{(u_i{v}_j)\in E(G_1)}{d}_{G_1\oplus G_2}(u_i,z_k)d_{G_1\oplus G_2}(v_j,w_l)\hfill \\ \hfill & \hfill & -4\sum _{(u_i{v}_j)\in E(G_1)}\sum _{(a_i{b}_j)\in E(G_2)}{d}_{G_1\oplus G_2}(u_i{a}_i)d_{G_1\oplus G_2}(v_j{b}_j)\hfill \\ \hfill & =\hfill & \sum _{x_i\in V(G_1)}\sum _{y_j\in V(G_1)}\sum _{(a_i{b}_j)\in E(G_2)}[(n_2+n_1{d}_{a_i}-2d_{x_i}{d}_{a_i})\hfill \\ \hfill & \hfill & (n_2{d}_{y_j}+n_1{d}_{b_j}-2d_{y_j}{d}_{b_j})]\hfill \\ \hfill & \hfill & +\sum _{z_k\in V(G_2)}\sum _{w_l\in V(G_2)}\sum _{(u_i{v}_j)\in E(G_1)}[(n_2{d}_{u_i}+n_1{d}_{z_k}-2d_{u_i}{d}_{z_k})\hfill \\ \hfill & \hfill & (n_2{d}_{v_j}+n_1{d}_{w_l}-2d_{v_j}{d}_{w_l})]\hfill \end{array}\\ \begin{array}{ccc}\hfill & \hfill & -4\sum _{(u_i{v}_j)\in E(G_1)}\sum _{(a_i{b}_j)\in E(G_2)}[(n_2{d}_{u_i}+n_1{d}_{a_i}-2d_{u_i}{d}_{a_i})\hfill \\ \hfill & \hfill & (n_2{d}_{v_j}+n_1{d}_{b_j}-2d_{v_j}{d}_{b_j})]\hfill \\ \hfill & =\hfill & \sum _{x_i\in V(G_1)}\sum _{y_j\in V(G_1)}\sum _{(a_i{b}_j)\in E(G_2)}(n_2^2{d}_{x_i}{d}_{y_j}+n_1{n}_2{d}_{x_i}{d}_{b_j}\hfill \\ \hfill & \hfill & -2n_2{d}_{x_i}{d}_{y_j}{d}_{b_j}+n_1{n}_2{d}_{a_i}{d}_{y_j}+n_1^2{d}_{a_i}{d}_{b_j}\hfill \\ \hfill & \hfill & -2n_1{d}_{a_i}{d}_{y_j}{d}_{b_j}-2n_2{d}_{x_i}{d}_{y_j}{d}_{a_i}\hfill \\ \hfill & \hfill & -2n_1{d}_{x_i}{d}_{a_i}{d}_{b_j}+4d_{x_i}{d}_{y_j}{d}_{a_i}{d}_{b_j})\hfill \\ \hfill & \hfill & +\sum _{z_k\in V(G_2)}\sum _{w_l\in V(G_2)}\sum _{(u_i{v}_j)\in E(G_1)}(n_2^2{d}_{u_i}{d}_{v_j}+n_1{n}_2{d}_{u_i}{d}_{w_l}\hfill \\ \hfill & \hfill & -2n_2{d}_{u_i}{d}_{v_j}{d}_{w_l}+n_1{n}_2{d}_{z_k}{d}_{v_j}+n_1^2{d}_{z_k}{d}_{w_l}\hfill \\ \hfill & \hfill & -2n_1{d}_{z_k}{d}_{v_j}{d}_{w_l}-2n_2{d}_{u_i}{d}_{z_k}{d}_{v_j}-2n_1{d}_{u_i}{d}_{z_k}{d}_{w_l}\hfill \\ \hfill & \hfill & +4d_{u_i}{d}_{v_j}{d}_{z_k}{d}_{w_l})\hfill \end{array}\end{array}$$ $$\begin{array}{c}\begin{array}{ccc}\hfill & \hfill & -4\sum _{(u_i{v}_j)\in E(G_1)}\sum _{(a_i{b}_j)\in E(G_2)}(n_2^2{d}_{u_i}{d}_{v_j}+n_1{n}_2{d}_{u_i}{d}_{b_j}\hfill \\ \hfill & \hfill & -2n_2{d}_{u_i}{d}_{v_j}{d}_{b_j}+n_1{n}_2{d}_{a_i}{d}_{v_j}+n_1^2{d}_{a_i}{d}_{b_j}\hfill \\ \hfill & \hfill & -2n_1{d}_{a_i}{d}_{v_j}{d}_{b_j}-2n_2{d}_{u_i}{d}_{v_j}{d}_{a_i}\hfill \\ \hfill & \hfill & -2n_1{d}_{u_i}{d}_{a_i}{d}_{b_j}+4d_{u_i}{d}_{v_j}{d}_{a_i}{d}_{b_j})\hfill \\ \hfill & =\hfill & 4n_2^2{m}_1^2{m}_2+\frac{n_1{n}_2}{2}\hfill \\ \hfill & \hfill & \left\{\sum _{x_i\in V(G_1)}\sum _{y_j\in V(G_1)}\sum _{(a_i{b}_j)\in E(G_2)}\mathrm{}\mathrm{}{d}_{x_i}(d_{a_i}\mathrm{}+\mathrm{}{d}_{b_j})\mathrm{}+\mathrm{}{d}_{y_j}(d_{a_i}\mathrm{}+\mathrm{}{d}_{b_j})\mathrm{}\right\}\hfill \\ \hfill & \hfill & -8m_1^2{n}_2{M}_1(S_2)+M_2(S_2)(n_1^4-8n_1^2{m}_1+16m_1^2)\hfill \\ \hfill & \hfill & +M_2(S_1)(n_2^4-8n_2^2{m}_2+16m_2^2)\hfill \\ \hfill & \hfill & +\frac{n_1{n}_2}{2}\{\sum _{z_k\in V(G_2)}\sum _{w_l\in V(G_2)}\sum _{(u_i{v}_j)\in E(G_1)}{d}_{z_k}(d_{u_i}+d_{v_j})\hfill \end{array}\\ \begin{array}{ccc}\hfill & \hfill & +d_{w_l}(d_{u_i}+d_{v_j})\}+4n_1^2{m}_2^2{m}_1+8n_1{m}_1^2{M}_1(S_1)\hfill \\ \hfill & \hfill & -8n_2{M}_2(S_1)M_1(S_2)-16M_2(S_1)M_2(S_2)\hfill \\ \hfill & \hfill & -8n_1{M}_1(S_1)M_2(S_2)+2n_1{n}_2{M}_1(S_1)M_1(S_2)\hfill \\ \hfill & =\hfill & 4n_2^2{m}_1^2{m}_2+\frac{4m_1{n}_1^2{n}_2}{2}{M}_1(S_2)-8m_1^2{n}_2{M}_1(S_2)+M_2(S_2)\hfill \\ \hfill & \hfill & (n_1^4-8n_1^2{m}_1+16m_1^2)+M_2(S_1)(n_2^4-8n_2^2{m}_2+16m_2^2)\hfill \\ \hfill & \hfill & +\frac{4m_2{n}_1{n}_2^2}{2}{M}_1(S_1)+8n_1{m}_1^2{M}_1(S_1)+4n_1^2{m}_2^2{m}_1\hfill \\ \hfill & \hfill & -8n_2{M}_2(S_1)M_1(S_2)+8n_1{m}_1^2{M}_1(S_1)+4n_1^2{m}_2^2{m}_1\hfill \\ \hfill & \hfill & -16M_2(S_1)M_2(S_2)-8n_1{M}_1(S_1)M_2(S_2)\hfill \\ \hfill & \hfill & +2n_1{n}_2{M}_1(S_1)M_1(S_2),\hfill \end{array}\end{array}$$which gives Equation (17)(17) $$\begin{array}{c}{M}_2(G_1\oplus G_2)=4n_2^2{m}_1^2{m}_2+4n_1^2{m}_2^2{m}_1\\ +(2m_1{n}_1^2{n}_2-8m_1^2{n}_2)M_1(S_2)\\ +(n_1^4-8n_1^2{m}_1+16m_1^2)M_2(S_2)\\ +(n_2^4-8n_2^2{m}_2+16m_2^2)M_2(S_1)\\ +(2m_2{n}_1{n}_2^2+8n_1{m}_2^2)M_1(S_1)\\ +2n_1{n}_2{M}_1(S_1)M_1(S_2)\\ -8n_2{M}_2(S_1)M_1(S_2)-8n_1{M}_1(S_1)M_2(S_2)\\ -16M_2(S_1)M_2(S_2),\end{array}$$(17) . □

3 Conclusion

The general expressions for the first and second Zagreb indices of five product graphs in terms of the Zagreb indices of their signed graphs and other graph parameters have been deduced in this study. The purpose of this article is to explore the mathematical properties of these indices but not their applications in chemical graph theory. Future scope is the computation of other degree-based or neighborhood degree sum-based topological indices for these product graphs in terms of the indices of their signed graphs.

Acknowledgments

The first author is highly obliged to TMA Pai Foundation for extending research fellowship vide, letter No. 209/SMU/REG/TMAPURF/70/2020 dated 30th September 2021.

Disclosure statement

The authors confirm there were no competing interests in the article content.