On the Signed 22-independence Number of Graphs

In this paper, we study the signed 2-independence number in graphs and give new sharp upper and lower bounds on the signed 2-independence number of a graph by a simple uniform approach. In this way, we can improve and generalize some known results in this area

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 2-independence function on G. 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 α2s(G) of G. In this work, we mainly present upper bounds on α2s(G), as for example α2s(G) ≤ n−2 [∆ (G)/2], and we prove the Nordhaus-Gaddum type inequality α2s (G) + α2s(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

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 2-independence function on $G$. 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 $α_s^2(G)$ of $G$. In this work, we mainly present upper bounds on $α_s^2(G)$, as for example $α_s^2(G) ≤ n−2 [∆ (G)//2]$, and we prove the Nordhaus-Gaddum type inequality $α_s^2 (G) + α_s^2(G) ≤ n+1$, 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