Graph labelling is one of the fascinating areas of graph theory with wide-ranging applications. Labelled graphs serve as useful models for a broad range of applications, such as X-ray, crystallography, radar, coding theory, astronomy, circuit design, channel assignments of FM radio stations and communication network addressing. Motivated by the problem of channel assignment of FM radio stations, the concept of radio labelling of graph and the notion of radio mean labelling of graph have been introduced. Motivated by the notion of radio mean labelling and its noteworthy results, the concept of radio harmonic mean labelling has been introduced. In this work, most of the results focus on particular classes of star graphs and utilise ad hoc methods. The methodology is used in this study, the vertices are assigned the labels with N, finding the span of a mapping f and finding the lowest spans taken over all radio harmonic mean labeling of the graph G.A radio harmonic mean labeling of a connected graph G is a one to one map f from the vertex set V(G) to the set of natural numbers N such that for any two distinct vertices u and v of G, \(d(u, v)+\left\lceil\frac{2 f(u) f(v)}{f(u)+f(v)}\right\rceil \geq 1+\operatorname{diam}(G)\). The radio harmonic mean number of f, rhmn(f) is the maximum number assigned to any vertex of G. The radio harmonic mean number of G, rhmn(G) is the minimum value of rhmn(f) taken over all radio harmonic mean labelling f of G. In this paper, we have determined the radio harmonic mean number of some star-related graphs.
Author(s) Details
R. Revathy
Department of Mathematics, Shri Shankarlal Sundarbai Shasun Jain College
for Women, T. Nagar, Chennai, Tamil Nadu, India.
K. Amuthavalli
Department of Mathematics, Government Arts and Science College,
Veppanthattai, Perambalur, Tamil Nadu, Bharathidasan University, India.
