Suppose each edge of a simple connected undirected Graph is given a unique number from the numbers $1, 2, \dots, $q$, where $q$ is the number of edges of that Graph. Then each vertex is labelled with sum of the labels of the edges incident to it. If no two vertices have the same label, then the Graph is called an antimagic Graph. We prove that the Cartesian product of wheel Graph and path Graph is antimagic.

Graph with vertex set V (G)  V (H) and u = (u1; v1) is adjacent with v = (u2; v2) whenever (u1 = u2 and v1 is adjacent with v2) or (v1 6= v2 and u1 is adjacent with u2). In this paper, we compute the exact values of the Wiener, vertex PI and Zagreb indices of deleted lexicoGraphic product of Graphs. Applications of our results under some examples are presented.

In this paper, we introduce a suitable generalization of Cayley Graphs that is de ned over polygroups (GCP-Graph) and give some examples and properties. Then, we mention a generalization of NEPS that contains some known Graph operations and apply to GCP-Graphs. Finally, we prove that the product of GCP-Graphs is again a GCP-Graph.

The Indu-Bala product of Graphs G and H consists of two disjoint copies of the join of G and H such that there is an  adjacency between the corresponding vertices in the two copies of H. A vertex subset S of a Graph G = (V, E) is said to be a geodetic set if every vertex in G is in some u−v geodesic, where u and v are any two vertices in S. The minimum cardinality of such a set is the geodetic number of G. The vertex subset D of a Graph G is said to be a dominating set if every vertex in G is either in D or adjacent to at least one vertex in D. The minimum cardinality of such a set is the domination number of G. In this work, the authors studied various geodetic and dominating extensions with respect to the Indu-Bala product of Graphs. The Aα matrix associated with a Graph is a convex linear combination of its adjacency matrix and degree diagonal matrix, offering deeper insights into the properties of both matrices. In this article the authors discuss the Aα spectrum of Indu-Bala product of Graphs.

Let $\mathcal{B}$ be a commutative ring with unity $1\neq 0$, $1\leq m <\infty$ be an integer and $\mathcal{R}=\mathcal{B}\times \mathcal{B} \times\cdots\times \mathcal{B}$ ($m$ times). The total essential dot product Graph $ETD(\mathcal{R})$ and the essential zero-divisor dot product Graph $EZD(\mathcal{R})$ are undirected Graphs with the vertex sets $\mathcal{R}^{*} = \mathcal{R}\setminus \{(0,0,...0)\}$ and $Z(\mathcal{R})^*=Z(\mathcal{R})\setminus \{(0,0,...,0)\}$ respectively. Two distinct vertices $w=(w_1,w_2,...,w_m)$ and $z=(z_1,z_2,...,z_m)$ are adjacent if and only if $ann_\mathcal{B}(w\cdot z)$ is an essential ideal of $\mathcal{B}$ (where $w\cdot z=w_1z_1+w_2z_2+\cdots +w_mz_m\in \mathcal{B}$). In this paper, we prove some results on connectedness, diameter and girth of $ETD(\mathcal{R})$ and $EZD(\mathcal{R})$. We classify the ring $\mathcal{R}$ such that $EZD(\mathcal{R})$ and $ETD(\mathcal{R})$ are planar, outerplanar, and of genus one.

Let $\mathscr{B}$ be a commutative ring with $1\neq 0$, $1\leq m<\infty$ be an integer and $\mathcal{R}=\mathscr{B}\times \mathscr{B}\times \cdot \cdot \cdot \times \mathscr{B}$ ($m$ times). In this paper, we introduce two types of (undirected) Graphs, total nilpotent dot product Graph denoted by $\mathcal{T_{N}D(\mathcal{R})}$ and nilpotent dot product Graph denoted by $\mathcal{Z_ND(\mathcal{R})}$, in which vertices are from $\mathcal{R}^\ast = \mathcal{R}\setminus \{(0,0,...,0)\}$ and $\mathcal{Z_{N}(\mathcal{R})}^*$ respectively, where $\mathcal{Z_{N}(\mathcal{R})}^{*}=\{w\in \mathcal{R}^*| wz\in \mathcal{N(R)}, \mbox{for some }z\in \mathcal{R}^*\} $. Two distinct vertices $w=(w_1,w_2,...,w_m)$ and $z=(z_1,z_2,...,z_m)$ are said to be adjacent if and only if $w\cdot z\in \mathcal{N}(\mathscr{B})$ (where $w\cdot z=w_1z_1+\cdots+w_mz_m$, denotes the normal dot product and $\mathcal{N}(\mathscr{B})$ is the set of nilpotent elements of $\mathscr{B}$). We study about connectedness, diameter and girth of the Graphs $\mathcal{T_ND(R)}$ and $\mathcal{Z_ND(R)}$. Finally, we establish the relationship between $\mathcal{T_ND(R)}$, $\mathcal{Z_ND(R)}$, $\mathcal{TD(R)}$ and $\mathcal{ZD(R)}$.

Neighbor-distinguishing colorings, which are colorings that induce a proper vertex coloring of a Graph, have been the focus of different studies in Graph theory. One such coloring is the set coloring. For a nontrivial Graph $G$, let $c:V(G)\to \mathbb{N}$ and define the neighborhood color set $NC(v)$ of each vertex $v$ as the set containing the colors of all neighbors of $v$. The coloring $c$ is called a set coloring if $NC(u)\neq NC(v)$ for every pair of adjacent vertices $u$ and $v$ of $G$. The minimum number of colors required in a set coloring is called the set chromatic number of $G$ and is denoted by $\chi_s (G)$. In recent years, set colorings have been studied with respect to different Graph operations such as join, comb product, middle Graph, and total Graph. Continuing the theme of these previous works, we aim to investigate set colorings of the Cartesian product of Graphs. In this work, we investigate the gap given by $\max\{ \chi_s(G), \chi_s(H) \} - \chi_s(G\ \square\ H)$ for Graphs $G$ and $H$. In relation to this objective, we determine the set chromatic numbers of the Cartesian product of some Graph families.

Let $\mathscr{B}$ be a commutative ring with $1\neq 0$, $1\leq m<\infty$ be an integer and $\mathcal{R}=\mathscr{B}\times \mathscr{B}\times \cdot \cdot \cdot \times \mathscr{B}$ ($m$ times). In this paper, we introduce two types of (undirected) Graphs, total nilpotent dot product Graph denoted by $\mathcal{T_{N}D(\mathcal{R})}$ and nilpotent dot product Graph denoted by $\mathcal{Z_ND(\mathcal{R})}$, in which vertices are from $\mathcal{R}^\ast = \mathcal{R}\setminus \{(0,0,...,0)\}$ and $\mathcal{Z_{N}(\mathcal{R})}^*$ respectively, where $\mathcal{Z_{N}(\mathcal{R})}^{*}=\{w\in \mathcal{R}^*| wz\in \mathcal{N(R)}, \mbox{for some }z\in \mathcal{R}^*\} $. Two distinct vertices $w=(w_1,w_2,...,w_m)$ and $z=(z_1,z_2,...,z_m)$ are said to be adjacent if and only if $w\cdot z\in \mathcal{N}(\mathscr{B})$ (where $w\cdot z=w_1z_1+\cdots+w_mz_m$, denotes the normal dot product and $\mathcal{N}(\mathscr{B})$ is the set of nilpotent elements of $\mathscr{B}$). We study about connectedness, diameter and girth of the Graphs $\mathcal{T_ND(R)}$ and $\mathcal{Z_ND(R)}$. Finally, we establish the relationship between $\mathcal{T_ND(R)}$, $\mathcal{Z_ND(R)}$, $\mathcal{TD(R)}$ and $\mathcal{ZD(R)}$.