Weakly (k,d)-arithmetic graph

Weakly [math]\displaystyle{ (k,d) }[/math]-arithmetic graph --- слабо [math]\displaystyle{ (k,d) }[/math]-арифметический граф.

A weakly arithmetic vertex function of a graph [math]\displaystyle{ G = (V,E) }[/math] is a vertex function [math]\displaystyle{ f: V(G) \rightarrow \{0,1,2, \ldots\} }[/math], such that, for specified positive integers [math]\displaystyle{ k }[/math] and [math]\displaystyle{ d }[/math], [math]\displaystyle{ \{k, k+d, k + 2d, \ldots\} }[/math] is the set of values of the induced edge function [math]\displaystyle{ f^{\ast} }[/math] defined by [math]\displaystyle{ f^{\ast}(uv) = f(u) + f(v) }[/math] for each edge [math]\displaystyle{ uv \in E(G) }[/math]. If a graph admits such a vertex function, [math]\displaystyle{ f }[/math], then [math]\displaystyle{ G }[/math] is said to be weakly ([math]\displaystyle{ k,d }[/math])-arithmetic.

In the above definition, if we impose the condition that the vertex function [math]\displaystyle{ f }[/math] is injective, then [math]\displaystyle{ f }[/math] is called a [math]\displaystyle{ (k,d) }[/math]-arithmetic num\-ber\-ing of the graph [math]\displaystyle{ G }[/math] and if a graph [math]\displaystyle{ G }[/math] admits such a numbering, then the graph [math]\displaystyle{ G }[/math] is called a [math]\displaystyle{ (k,d) }[/math]-arithmetic graph.