For a graph G = (V, E) and a coloring f : V (G) → Z 2 let vf (i) = |f−1(i)|. f is said to be friendly if |vf (1)−vf (0)| ≤ 1. The coloring f : V (G) → Z 2 induces an edge labeling f∗ : E(G) → Z 2 defined by f∗(xy) = f(x) + f(y) ∀xy ∈ E(G), where the summation is done in Z 2. Let ef (i) = |f∗−1(i)|. The friendly index set of the graph G, denoted by FI(G), is defined by FI(G) = {|ef (1)− ef (0)| ...

For a graph G = (V,E) and a binary labeling (coloring) f : V (G) → Z2, let vf (i) = |f−1(i)|. f is said to be friendly if |vf (1) − vf (0)| ≤ 1. The labeling f : V (G) → Z2 induces an edge labeling f∗ : E(G) → Z2 defined by f∗(xy) = |f(x)− f(y)| ∀xy ∈ E(G). Let ef (i) = |f∗−1(i)|. The friendly index set of the graph G, denoted by FI(G), is defined by FI(G) = {|ef (1)− ef (0)| : f is a friendly ...

Let Ω be an algebraic closure of Qp and let F be a finite extension of Qp contained in Ω. Given positive integers f and e, the number of extensions K/F contained in Ω with residue degree f and ramification index e was computed by Krasner. This paper is concerned with the number I(F, f, e) of F -isomorphism classes of such extensions. We determine I(F, f, e) completely when p2 ∤ e and get partia...

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...