Abstract:
A radio k-labeling c of a graph G is a mapping c : V (G) → Z+ ∪ {0}, such that
d(x, y) + |c(x) − c(y)| ≥ k + 1
holds for every two distinct vertices x and y of G, where d(x, y) is the distance between
any two vertices x and y of G. The span of a radio k-labeling c is denoted by sp(c)
and defined as max{|c(x) − c(y)| : x, y ∈ V (G)}. The radio labeling is a radio klabeling
when k = diam(G). In other words, a radio labeling is a one-to-one function
c : V (G) → Z+ ∪ {0}, such that
|c(x) − c(y)| ≥ diam(G) + 1 − d(x, y)
for any pair of vertices x, y in G. The radio number of G denoted by rn(G), is the lowest
span taken over all radio labelings of the graph. When k = diam(G) − 1, a radio klabeling
is called a radio antipodal labeling. An antipodal labeling for a graph G is a
function c : V (G) → {0, 1, 2, ...}, so that
d(x, y) + |c(x) − c(y)| ≥ diam(G)
for all x, y ∈ G. The radio antipodal number for G denoted by an(G), is the minimum
span of an antipodal labeling admitted by G. In this thesis, we investigate the exact value
of the radio number and radio antipodal number for different family of graphs. Further
more, we also determine the lower bound of the radio number for some cases.
