An injective function F  where, F : V(G) → {0,2,4, ..., 2b + 2d - 2}  is said to be d-even vertex odd mean labeling (d-EVOML) of the graph G(a, b)  when the induced mapping F* : E(G) → {1,3, ..., 2b - 1}  given by: F*(lw) = F(l)+F(w)/2, is a bijective function. A d - even vertex odd mean graph is a graph which allows even vertex odd mean labelling. In this study, we identify the lowest value of d for which the graphs: Y-tree, star graph, Px ʘ Kh,  crown graph Rh,  and rooted product Ph◊C4  have a d- even vertex odd mean labeling. Furthermore, we find the minimum number d for which the graphs: cycle graph Ch when h ≡ 2 mod 4,  dragon graph Px(Ch)  when h ≡ 2 mod 4, x ≥ 1,  prism graph Ph,  and Toeplitz graphs Th(1, 3), Th(1, 5)  and Th(1, 3, 5)  have a similar d- even vertex odd mean labeling. In the end, we establish that no odd cycle Ch  is an even vertex odd mean graph for all d.