On γ-labelings of trees

Abstract

Let G be a graph of order n and size m. A γ-labeling of G is a oneto-one function f : V (G) → {0, 1, 2,. .. , m} that induces a labeling f : E(G) → {1, 2,. .. , m} of the edges of G defined by f (e) = |f (u)−f (v)| for each edge e = uv of G. The value of a γ-labeling f is val(f) = e∈E(G) f (e). The maximum value of a γ-labeling of G is defined as val max (G) = max{val(f) : f is a γ-labeling of G}; while the minimum value of a γ-labeling of G is val min (G) = min{val(f) : f is a γ-labeling of G}.

THE gamma-SPECTRUM OF CYCLE WITH ONE CHORD

“…The γ-spectra of stars, paths, cycles and complete graphs are determined in [2,7]. Next we recall the γ-spectrum of C n for each integer n ≥ 3.…”

“…In 2005 Chartrand, Erwin, VanderJagt, and Zhang [2] studied, for a graph G of order n and size m, a γ-labeling of G defined as a one-to-one function f : V (G) → {0, 1, 2, . .…”

“…The maximum value and the minimum value of a γ-labeling of G are defined in [2] as val max (G) = max{val(f ) : f is a γ-labeling of G} and val min (G) = min{val(f ) : f is a γ-labeling of G} , respectively. It turns out that val max (P 5 ) = 11 and val min (P 5 ) = 4.…”

“…The subject of maximum and minimum values of a γ-labeling of path P n , cycle C n , complete graph K n , double star S p,q and complete bipartite graph K r,s and cycle with a triangle C △ n are studied in [1,2,3,5,6]. Later, the extreme values of γ-labeling of cycle with one chord C n + e, i.e., cycle with a chord e joining two nonadjacent vertices in cycle C n , are established in [10] as we state next.…”

Radio Labelings and Antipodal Labelings of Trees with Order Up to Eight

“…For more information of the study of radio labeling, please consult. [7][8][9] In this paper, we establish an integer linear programming model and a Satisfiability model to solve the radio k-labeling instances. With a SAT solver or an integer linear programming solver, we are able to determine all the radio numbers and antipodal numbers of trees with order up to eight.…”