On balance index sets of generalized book and ear expansion graphs

Let G be a simple graph with vertex set V(G) and edge set E(G), and let Z2 = {0,1}. A labeling f : V(G) → Z2 induces a partial edge labeling f*: E(G) Z2 defined by f*(uv) =f(u) if and only if f(u) = f(v). For i Z2, let vf(i) ={v V(G) : f(v) = i} and ef(i) = |{e E(G) : f*(e) = i}. A labeling f is called a friendly labeling if |v f(0) - vf(l)| ≤ 1. The BI(G), the balance index set of G, is defined as {|ef(0) - ef(l)| : the vertex labeling f is friendly.}. This paper focuses on the balance index sets of generalized book and ear expansion graphs.