Bounds on the signed 2-independence number in graphs

Let G be a finite and simple graph with vertex set V (G), and let f : V (G) → {−1, 1} be a two-valued function. If ∑ x∈N [v] f(x) ≤ 1 for each v ∈ V (G), where N [v] is the closed neighborhood of v, then f is a signed 2independence function onG. The weight of a signed 2-independence function f is w(f) = ∑ v∈V (G) f(v). The maximum of weights w(f), taken over all signed 2-independence functions f on G, is the signed 2-independence number α s(G) of G. In this work, we mainly present upper bounds on α s(G), as for example α s(G) ≤ n−2⌈∆(G)/2⌉, and we prove the Nordhaus-Gaddum type inequality α s(G) + α 2 s(G) ≤ n+ 1, where n is the order and ∆(G) is the maximum degree of the graph G. Some of our theorems improve well-known results on the signed 2-independence number.