ADV MATH SCI JOURNAL Advances in Mathematics: Scientific Journal 9 (2020), no.11, 9443–9453 ISSN: 1857-8365 (printed); 1857-8438 (electronic) https://doi.org/10.37418/amsj.9.11.48 ZERO FORCING GRAPH ASSOCIATED TO THE TOTAL GRAPH OF Z n WITH RESPECT TO NIL IDEAL ARIJIT MISHRA 1 AND KUNTALA PATRA ABSTRACT. The total graph T (Γ N (Z n )) of Z n with respect to its nil ideal N (Z n )= {x ∈ Z n : x 2 ≡ 0(mod n)} is a simple, undirected graph with vertex set Z n and any two distinct vertices x and y of T (Γ N (Z n )) are adjacent if and only if x + y ∈ N (Z n ). In this paper, we introduce a new graph structure called a Zero forcing graph of T (Γ N (Z n )), denoted by ZF (T (Γ N (Z n ))), as a simple, undi- rected graph in which all the possible zero forcing sets of minimum cardinality of T (Γ N (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 ∪ S 2 = Z n . 1. I NTRODUCTION The idea of the total graph of a commutative ring R, denoted by T (Γ(R)), was first put forward by Anderson and Badawi [7] who defined it as a simple, undirected graph with vertex set R and any two distinct vertices x and y are adjacent if and only if x + y ∈ Z (R), where Z (R) denotes the set of all the zero-divisors of R. In the year 2003, P. W. Chen [12] introduced a new class of a graph of a commutative ring R with vertex set R and two distinct vertices x and y are adjacent if and only if xy ∈ N (R), where N (R) denotes the set of all the nil elements of the ring R. This concept was further modified by Ai-Hua Li and 1 corresponding author 2020 Mathematics Subject Classification. 05C25, 05C69. Key words and phrases. Total Graph, Nil Ideal, Zero Forcing Set, Zero Forcing Number. 9443