ZERO FORCING GRAPH ASSOCIATED TO THE TOTAL GRAPH OF Zn WITH RESPECT TO NIL IDEAL

A. Mishra and K. Patra

  DOI:  https://doi.org/10.37418/amsj.9.11.48

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The total graph $T(\Gamma_N(\mathbb{Z}_n))$ of $\mathbb{Z}_n$ with respect to its nil ideal $N(\mathbb{Z}_n)= \lbrace x \in \mathbb{Z}_n : x^2 \equiv 0 ($mod $n) \rbrace$ is a simple, undirected graph with vertex set $\mathbb{Z}_n$ and any two distinct vertices $x$ and $y$ of $T(\Gamma_N(\mathbb{Z}_n))$ are adjacent if and only if $x+y \in N(\mathbb{Z}_n)$. In this paper, we introduce a new graph structure called a \textit{Zero forcing graph} of $T(\Gamma_N(\mathbb{Z}_n))$, denoted by $\mathcal{ZF}(T(\Gamma_N(\mathbb{Z}_n)))$, as a simple, undirected graph in which all the possible zero forcing sets of minimum cardinality of $T(\Gamma_N(\mathbb{Z}_n))$ are taken as vertices and any two distinct vertices $S_1$ and $S_1$ of this graph are adjacent if and only if $S_1 \cup S_2 =\mathbb{Z}_n$. \\

Keywords: Total Graph, Nil Ideal, Zero Forcing Set, Zero Forcing Number.