Menu
Let \(G\) be a graph with vertex set \(V(G)\) and edge set \(E(G)\). A labeling \(f\) of a graph \(G\) is said to be edge-friendly if \(|e_f(0) – e_f(1)| \leq 1\), where \(e_f(i) = \text{card}\{e \in E(G) : f(e) = i\}\). An edge-friendly labeling \(f : E(G) \to \mathbb{Z}_2\) induces a partial vertex labeling \(f^+ : V(G) \to A\) defined by \(f^+(x) = 0\) if the edges incident to \(x\) are labeled \(0\) more than \(1\). Similarly, \(f^+(x) = 1\) if the edges incident to \(x\) are labeled \(1\) more than \(0\). \(f^+(x)\) is not defined if the edges incident to \(x\) are labeled \(1\) and \(0\) equally. The edge-balance index set of the graph \(G\), \(EBI(G)\), is defined as \(\{|v_f(0) – v_f(1)| : \text{the edge labeling } f \text{ is edge-friendly}\}\), where \(v_f(i) = \text{card}\{v \in V(G) : f^+(v) = i\}\).
An \(n\)-wheel is a graph consisting of \(n\) cycles, with each vertex of the cycles connected to one central hub vertex. The edge-balance index sets of \(n\)-wheels are presented.