The integral circulant graph \(\textrm{ICG}_n (D)\) has the vertex set \(Z_n = \{0, 1, 2, \ldots , n - 1\}\) , where vertices a and b are adjacent if \(\gcd (a-b,n)\in D\) , with \(D \subseteq \{d: d \mid n,\ 1\le d<n\}\) . In this paper, we establish that the minimal value of the least eigenvalues (minimal least eigenvalue) of integral circulant graphs \(\textrm{ICG}_n(D)\) , given an order n with its prime factorization \(p_1^{\alpha _1}\cdots p_k^{\alpha _k}\) , is equal to \(-\frac{n}{p_1}\) . Moreover, we show that the minimal least eigenvalue of connected integral circulant graphs \(\textrm{ICG}_n(D)\) of order n whose complements are also connected is equal to \(-\frac{n}{p_1}+p_1^{\alpha _1-1}\) . Finally, we determine the second minimal eigenvalue among all least eigenvalues within the class of connected integral circulant graphs of a prescribed order n and show it to be equal to \(-\frac{n}{p_1}+p_1-1\) or \(-\frac{n}{p_1}+1\) , depending on whether \(\alpha _1>1\) or not, respectively. In all the aforementioned tasks, we provide a complete characterization of graphs whose spectra contain these determined minimal least eigenvalues.
