On Some Basic Graph Invariants of Unitary Addition Cayley Graph of Gaussian Integers Modulo n

Graphs Defined on Rings: A Review

“…In [215][216][217], the properties of the unitary addition Cayley graph of the ring of Gaussian integers modulo n, Z n [i] (refer to Definition 16) were investigated, where the exact values and bounds of certain parameters of the graph Z n [i] were obtained. Note that the number of elements in the ring Z n [i] is n 2 , as there are n ways to fill both the real and the complex part of the number a + ib.…”

“…The degrees of the vertices, size, diameter, and girth of the unitary addition Cayley graph on Z n [i] was given in [215], based on the value of n, as mentioned in Theorem 97, from which it was determined that the unitary addition Cayley graph on Z n [i] is a complete bipartite graph if and only if n = 2 t , t ∈ N. The traversal properties of the graph were also investigated in [215]; it was proven that the unitary addition Cayley graph on Z n [i] is always Hamiltonian and, when n is even, the graph is Eulerian. It was also found that the unitary addition Cayley graph on Z n [i] is planar only for n = 1, 2.…”

“…Adding to the study, the basic graph invariants for the unitary addition Cayley graph on Z n [i] were computed in [216,217]. Some bounds for the chromatic and the clique number of the graph were given in [217] as well as [216], which coincided with each other.…”

“…Adding to the study, the basic graph invariants for the unitary addition Cayley graph on Z n [i] were computed in [216,217]. Some bounds for the chromatic and the clique number of the graph were given in [217] as well as [216], which coincided with each other. In [216], the clique covering number of the unitary addition Cayley graph on Z n [i] was determined by determining the independence number of its complement, and, in [217], the domination number of the graph was obtained as either 1, 2, or 3, based on the value of n. A similar study was conducted on the unitary addition Cayley graphs of the ring Einstein integers modulo n, Z e n [i] (refer to Definition 17), in [218], where, along with the basic properties and parameters of the unitary addition Cayley graphs of Z e n [i], a comparison between the unitary addition Cayley graphs of the rings Z n [i] and Z e n [i] was also given, to enable a better comprehension of the structure of the rings, graphs, and their properties.…”

“…Along with the characterisation of the signed graphs based on the above-mentioned four properties, the characterisations of these properties of balance, clusterability, etc. in certain derived signed graphs from the signed graphs, such as the negation of the signed graph, and some variations of line signed graphs were also investigated in [218,221,222].…”