On friendly index and product-cordial index sets of Mobius-liked graph

Let G be a simple graph with vertex set V(G) and edge set E(G). Let (Z(2),+,*) be a field with two elements. A vertex labeling f V(G) -> Z(2) induces two edge labelings f(+) : E(G) -> Z(2) such that f(+) (xy) = f(x) +f(y), whereas f(+) : E(G)-> Z(2) such that f* (xy) = f(x) f(y), for each edge xy is an element of E(G). For i is an element of Z(2), let v(f) (i) =vertical bar {v is an element of V(G) : f(v) = i} vertical bar,e(f)(+)(i)=vertical bar{e is an element of E(G):f(+)(e) = i}vertical bar and e(f)*(i) = vertical bar{e is an element of E(G): f*(e) =i}vertical bar. A labeling f of a graph G is said to be friendly if vertical bar v(f) (0) - v(f) (1) <= 1. The friendly index set of the graph G, denoted FI(G), is defined as { vertical bar e(f)(+)(1) - e(f)(+)(0) vertical bar: the vertex labeling f is friendly}. This is a generalization of graph cordiality. The corresponding multiplicative version is called the product-cordial index set, denoted PCI(G), defined as {vertical bar e(f)(+)(1) - e(f)(+)(0)vertical bar: the vertex labeling f is friendly}. in this paper, we investigate the friendly index and product-cordial index sets of a family of cubic graphs known as Mobius-liked graph, MG(n) for even n >= 4.