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

Figures & data

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.	$d_{(G_1\oplus G_2)}(u_1,u_2)=n_2{d}_{G_1}(u_1)+n_1{d}_{G_2}(u_2)-2d_{G_1}(u_1)d_{G_2}(u_2).$