On friendly index sets of cycles with parallel chords

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) + fey), for each edge xy e E(G). For i e A, let vt<i) = card { v e V(G) : f(v) = i} and er(i) = card ( e e E(G) : f"'(e) = i}. Let c(f) = {Iet<i) etG)1 : (i, j) e A x A}. A labeling f of a graph G is said to be A­ friendly if I vt<i) Vf(i) I I for all (i, j) e A x A. If c(f) is a (0, I)-matrix for an A-friendly labeling f, then f is said to be A-cordial. When A = 'hz, the.{riendly index set of the graph G, FI(G), is defined as {ler(O) et<I)1 : the vertex labeling f is 'hz-friendly}. In [13] the friendly index set of cycles are completely determined. In this paper we describe the friendly index sets of cycles with parallel chords. We show that for a cycle with an arbitrary non-empty set of parallel chords, the numbers in its friendly index set form an arithmetic progression with common difference 2.