The least eigenvalues of integral circulant graphs

The integral circulant graph ICG(n)(D) has the vertex set Z(n) = {0,1,2,& mldr;,n-1}, where vertices a and b are adjacent if gcd(a-b,n) is an element of D, with D subset of {d: d divided by n, 1 <= d < n}. or In this paper, we establish that the minimal value of the least eigenvalues (minimal least eigenvalue) of integral circulant graphs ICGn(D), given an ordern with its prime factorization p(1)(alpha 1)<middle dot> <middle dot> <middle dot> p(k)(alpha k), is equal to -n/p1. Moreover, we show that the minimal least eigenvalue of connected integral circulant graphs ICG(n)(D)of order n whose complements are also connected is equal to-n/p1 + 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-n/p(1) + p(1)-1 or -n/p(1)+1, depending on whether alpha(1 )>1 not, respectively. In all the afore mentioned tasks, we provide a complete characterization of graphs whose spectra contain these determined minimal least eigenvalues.