Combinatorial Problems on Algebraic Configured Graphs

Abstract

Zero divisor graph and unitary addition Cayley graph are the targeted algebraic networks. Our first objective is to allocate the frequencies to their every channel (nodes) at the same time reduce the consumption of their spectral domain, in this case upper frequency bound is denoted by \(\lambda\). Next, decycle the aimed networks in order to reconstruct as trees or forest which has the less proximity than others. Let \(\varphi\)(\(\Gamma\)UA(Rn)) denotes the decycling number of \(\Gamma\)UA(Rn) which is the cardinality of the smallest decycling set of the unitary addition Cayley graph \(\Gamma\)UA(Rn) and A(\(\Gamma\)UA(Rn); x) denotes the acyclic polynomial of \(\Gamma\)UA(Rn). In this chapter, we obtained the frequency bound by the distance labeling constraint which is defined based on the diameter of a graph. Here, \(\lambda\) 2;1(\(\Gamma\)Rp ) and \(\lambda\) 3;2;1(\(\Gamma\)Rp ) denotes the least upper frequency bounds of zero divisor graphs. Also we obtained the acyclic number bounds and we derived the acyclic polynomial with their roots.