On the balance index set of the chain-sum graphs of cycles

Abstract

Let G be a graph with vertex set V(G) and edge set E(G), and let A ={0, 1}. A labeling f: V(G) → A induces a partial edge labeling f*: E(G) → A defined by f*(xy) = f(x) if and only if f(x) = f(y), for each edge xy ∈ E(G). For i ∈ A, let V_f(i) = card{v ∈ V(G): f(v) = i} and e_f(i) = card{e ∈ E(G): f*(e) = i}. A labeling f of a graph G is said to be friendly if | V_f(0) - V _f(1) | ≤ 1. If | e_f(0) - e_f(1) | ≤ 1 for a friendly labeling f, then G is said to be balanced. Balancedness of the trees is studied in [2]. The balance Index set of the graph G, BI(G), is defined as {| e_f(0) - e_f(1) : the vertex labeling f is friendly}. In this paper we determine balance index sets of some chain sum of graphs.