Further Results on Radio Number of Wedge sum of Graphs

Abstract

Let \(G\) be a simple connected graph with vertex set \(V\) and diameter \(d\). An injective function \(c: V\rightarrow \{1,2,3,\ldots\}\) is called a {radio labeling} of \(G\) if \({|c(x)-c(y)|+d(x,y)\geq d+1}\) for all distinct \(x,y\in V\), where \(d(x,y)\) is the distance between vertices \(x\) and \(y\). The largest number in the range of \(c\) is called the span of the labeling \(c\). The radio number of \(G\) is the minimum span taken over all radio labelings of \(G\). For a fixed vertex \(z\) of \(G\), the sequence \((l_1,l_2,\ldots,l_r)\) is called the level tuple of \(G\), where \(l_i\) is the number of vertices whose distance from \(z\) is \(i\). Let \(J^k(l_1,l_2,\ldots,l_r)\) be the wedge sum (i.e., one vertex union) of \(k\geq2\) graphs having same level tuple \((l_1,l_2,\ldots,l_r)\). Let \(J\left(\frac{l_1}{l'_1},\frac{l_2}{l'_2},\ldots,\frac{l_r}{l'_r}\right)\) be the wedge sum of two graphs of same order, having level tuples \((l_1,l_2,\ldots,l_r)\) and \((l'_1,l'_2,\ldots,l'_r)\). In this paper, we compute the radio number for some sub-families of \(J^k(l_1,l_2,\ldots,l_r)\) and \(J\left(\frac{l_1}{l'_1},\frac{l_2}{l'_2},\ldots,\frac{l_r}{l'_r}\right)\).