Order-Divisor Graphs of Γ1-Nonderanged Permutation Groups: Structure and Invariants

For each prime \(p \geq 5\), the \(\Gamma_1\)-nonderanged permutation group \(G_p^{\Gamma_1}=\left\{\omega_1, \ldots, \omega_{p-1}\right\}\), with multiplication \(\omega_i \omega_j=\omega_{i j \bmod p}\), is cyclic of order \(n=p-1\). Its order-divisor graph has the group elements as vertices, with distinct vertices adjacent precisely when their orders differ and one order divides the other. In view of the established divisor-class construction for cyclic groups, this graph is an independent-set expansion of the divisibility-comparability graph on the divisors of \(n\). We express this description in the permutation indexing, obtain formulas for the degrees and number of edges, and determine connectivity, regularity, diameter, radius, girth, center, and periphery. In particular, the identity is always universal; the unique involution is also universal exactly when \(p-1\) is a power of 2 . The graph is complete multipartite precisely when \(p-1=2^k(k \geq 2)\) or \(p-1=2 q\) for an odd prime \(q\). Explicit matrices and graph drawings for \(p=5,7,11\), together with direct enumeration for all 107 primes \(5 \leq p<600\), illustrate and check the specializations.