On Skolem odd and even difference mean graphs

Let G=(V,E) be a simple, finite and undirected (p,q)-graph with p vertices and q edges. A graph G is Skolem odd difference mean if there exists an injection f:V(G)→{0,1,2,…,p+3q-3} and an induced bijection f∗:E(G)→{1,3,5,…,2q-1} such that each edge uv (with f(u)>f(v)) is labeled with f∗(uv)=f(u)-f(v)2. We say G is Skolem even difference mean if there exists an injection f:V(G)→{0,1,2,…,p+3q-1} and an induced bijection f∗:E(G)→{2,4,6,…,2q} such that each edge uv (with f(u)>f(v)) is labeled with f∗(uv)=f(u)-f(v)2. A graph that admits a Skolem odd (or even) difference mean labeling is called a Skolem odd (or even) difference mean graph. In this paper, first, we construct some new Skolem odd difference mean graphs and then investigate the Skolem even difference meanness of some standard graphs.