Sum graph

Sum graph --- граф сумм.

A graph [math]\displaystyle{ G(V,E) }[/math] is called a sum graph, if there is an injective labeling called sum labeling [math]\displaystyle{ L }[/math] from [math]\displaystyle{ V }[/math] to a set of distinct positive integers [math]\displaystyle{ S }[/math], such that [math]\displaystyle{ xy \in E }[/math] if and only if there is a vertex [math]\displaystyle{ w }[/math] in [math]\displaystyle{ V }[/math] such that [math]\displaystyle{ L(w) = L(x) + L(y) \in S }[/math]. In this case [math]\displaystyle{ w }[/math] is called a working vertex. Every graph can be made into a sum graph by adding some isolated vertices, if necessary.

Sum graph labeling offers a new method for defining graphs and for storing them digitally.