Bounds on radio mean number of graphs

Abstract

 A radio mean labeling l of G maps distinct vertices of G to distinct elements of ℤ + satisfying the radio mean condition that diam ( G ) + 1 - d G ( w , w ′ ) ≤ ⌈ l ( w ) + l ( w ′ ) 2 ⌉ , ∀ w , w ′ ∈ V ( G ) where dG (w, w′) is the smallest length of a w, w′- path in G and diam (G) = max {dG (w, w′) : w, w′ ∈ V (G)} is the diameter of G. The radio mean number of graph G is defined as rmn (G) = min {span (l) : l isaradiomeanlabelingof G} where span(l) is given by max {l (w) : w ∈ V (G)}. It has been proved in literature that |V (G) | ≤ rmn (G) ≤ |V (G) | + diam (G) -2. Cryptographic algorithms can exploit the unique radio mean number associated with a graph to generate keys. An exhaustive listing of all feasible radio mean labelings and their span is essential to obtain the radio mean number of a given graph. Since the radio mean condition depends on the distance between vertices and the graph’s diameter, as the order and diameter increase, finding a radio mean labeling itself is quite difficult and, so is obtaining the radio mean number of a given graph. Here we discuss the extreme values of the radio mean number of a given graph of order n. In this article we obtained bounds on the radio mean number of a graph G of order n and diameter d in terms of the radio mean number of its induced subgraph H where diam (H) = d and dH (w, w′) = dG (w, w′) for any w, w′ ∈ V (H). The diametral path Pd+1 is one such induced subgraph of G and hence we have deduced the limits of rmn (G) in terms of rmn (Pd+1). It is known that if d = 1,  2 or 3, then rmn (G) = n. Here, we have given alternative proof for the same. The authors of this article have studied radio mean labeling of paths in another article. Using those results, we have improved the bounds on the radio mean number of a graph of order n and diameter d ≥ 4. It is also shown that among all connected graphs on n vertices, the path Pn of order n possesses the maximum radio mean number. This is the first article that has completely solved the question of maximum and minimum attainable radio mean numbers of graphs of order n.