Asian Journal of Pure and Applied Mathematics

A Γ₁-nonderanged permutation group, denoted by \(G^{\Gamma_1}_P\) for prime p\(\ge\)5, is defined as \(G^{\Gamma_1}_P\) = {\(\omega_i:1\le i \le p-1\)}  where each \(\omega_i\) is a permutation of the form \(\omega_i\) = (1) (1+i) mod p (1+2i) mod p ... (1+(p-1)i) mod p. This group has gained increasing attention due to its unique algebraic structure and its connection to non-deranged permutations. In this study, we investigate the undirected inverse graph representation of the Γ₁- nonderanged permutation group, denoted by inv(\(G^{\Gamma_1}_P\)). We show that the graph inv(\(G^{\Gamma_1}_P\)) is regular only when p = 5, and for every prime p \(\ge\) 5, both the diameter and radius of the graph are equal to 2. We further establish that the maximum degree of the graph is p-3 for all primes p \(\ge\) 5. Visual illustrations of the inverse graphs for selected prime values are presented, and a Python algorithm for generating inv(\(G^{\Gamma_1}_P\)) is provided.