A graph \(G(V,E)\) is declared to concede vertex prime and edge ternion sum labeling when the graph vertices are labeled with unique integers from \([1,|V|]\) so that for every edge \(m_i n_j\) , the end vertices \(m_i\) and \(n_j\) are allocated labels such that \(gcf(m_i,n_j )=1\) and the edges of the graph are assigned labels with distinct integers from \([1,|V|+|E|]\) such that the ternion sum of the designated labels of the vertices \(m_i,n_j\) and the edge \(m_i n_j\) is an unique constant. This unique constant is known as uniform ternion sum. In this research article, we have introduced 1-Edge varied ternion sum, k-Edge varied ternion sum, and k-edge varied ternion sum with l-variations, and we investigate that the centralized path network, centralized cycle network, centralized fan network, centralized triangular path network admit vertex prime and edge ternion sum labeling.