On Friendly Index Sets of Spiders
Sin-Min Lee, Ho-Kuen Ng, Gee-Choon Lau · 2014
Let G be a graph with vertex set V(G) and edge set E(G), and let A be an abelian group. A labeling f : V (G) → A induces an edge labeling f* : E(G) → A defined by f*( xy ) = f(x)+ f(y), for each edge xy ∈ E(G). For i ∈ A, let vf (i) = |{ v ∈ V(G) : f(v) = i}| and ef (i) = |{ e ∈ E(G) : f*(e) = i}|. Let c(f) = {| ef (i) � ef (j)| : ( i, j) ∈ A × A}. A labeling f of a graph G is said to be A-friendly if |vf (i) � vf (j)| ≤ 1 for all ( i, j) ∈ A × A. If c(f) is a (0, 1)-matrix for an A-friendly labeling f, then f is said to be A-cordial. When A = Z2, the friendly index set of the graph G, FI (G), is defined as {| ef (0) � ef (0)|: the vertex labeling f is Z2-friendly}. In this paper, we determined the friendly index sets of many spiders.