On a construction of quadruple circulant networks with the maximal number of nodes for any diameter

For undirected circulant networks, the maximization problem for the number of nodes under given degree and diameter of a graph is considered. A new lower estimate is obtained for the attainable number of nodes in the circulant graphs of dimension 4 and any diameter. Some new infinite families of circulants reaching this estimate are constructed. For graphs of these families, some analytical descriptions are given.