Frequency assignment model of zero divisor graph (Q2034700)

Summary: Given a frequency assignment network model is a zero divisor graph \(\Gamma= (V,E)\) of commutative ring \(R_\eta\), in this model, each node is considered to be a channel and their labelings are said to be the frequencies, which are assigned by the \(L(2,1)\) and \(L(3,2,1)\) labeling constraints. For a graph \(\Gamma\), \((L(2,1)\) labeling is a nonnegative real valued function \(f:V(G) \longrightarrow [0,\infty)\) such that \(\mid f(x)-f(y)\mid\geq 2d\) if \(d=1\) and \(\mid f(x)-f(y)\mid\geq d\) if \(d=2\) where \(x\) and \(y\) are any two vertices in \(V\) and \(d>0\) is a distance between \(x\) and \(y\). Similarly, one can extend this distance labeling terminology up to the diameter of a graph in order to enhance the channel clarity and to prevent the overlapping of signal produced with the minimum resource (frequency) provided. In general, this terminology is known as the \(L(h,k)\) labeling where \(h\) is the difference of any two vertex frequencies connected by a two length path. In this paper, our aim is to find the minimum spanning sharp upper frequency bound \(\lambda_{(2,1)}\) and \(\lambda_{(3,2,1)}\), within \(\Delta^2\), in terms of maximum and minimum degree of \(\Gamma\) by the distance labeling \(L(2,1)\) and \(L(3,2,1)\), respectively, for some order \(\eta= p^nq\), \(pqr\), \(p^n\) where \(p\), \(q\), \(r\) are distinct prime and \(n\) is any positive integer.