A graph G(V,E) is observed to admit Vertex Prime and Edge Ternion Sum Labeling when the vertices of the graph are labeled with unique integral values from [1,|V|] in a way that for each edge uv, the end vertices u and v are designated labels that share no common positive factors except 1 and the edges of the graph are assigned labels with distinct integral values from [1,|V|+|E|] such that the ternion sum of the designated labels of the vertices u,v and the edge uv constitutes uniform ternion sum or 1-edge varied ternion sum or k-edge varied ternion sum. In this research article we investigate that the uniform theta graph, quasi-uniform theta graph and subdivision of uniform theta graph admit Vertex Prime and Edge Ternion Sum Labeling.