On The Unitary Cayley Ring Signed Graphs$S_n^\oplus$

Abstract

A Signed graph (or sigraph in short) is an ordered pair S = (G, σ), where G is a graph G = (V, E) and σ : E → {+, −} is a function from the edge set E of G into the set {+, −}. For a positive integer n > 1, the unitary Cayley graph Xn is the graph whose vertex set is Zn, the integers modulo n and if Un denotes set of all units of the ring Zn, then two vertices a, b are adjacent if and only if a − b ∈ Un. In this paper, we have obtained a characterization of balanced and clusterable unitary Cayley ring sigraph [Formula: see text]. Further, we have established a characterization of canonically consistent unitary Cayley ring sigraph [Formula: see text], where n has at most two distinct odd primes factors. Also sign-compatibility has been worked out for the same.

Graphs Defined on Rings: A Review

“…The above-mentioned four variations of the signed unitary Cayley graphs were examined in [143,[145][146][147], where the properties of the unitary Cayley join signed graph and its negation were investigated in [146], the unitary Cayley ring signed graph was investigated in [145], and the unitary Cayley meet signed graph was explored in [143,147]. In [146], a characterisation of the balanced unitary Cayley join signed graphs and canonically consistent unitary Cayley join signed graphs S ∨ n , where n had at most two distinct odd prime factors, were obtained, as follows.…”

“…Unitary Cayley ring signed graphs, which are closely associated with unitary Cayley join signed graphs, were examined in [145]. It can be seen that an edge in a unitary Cayley join signed graph is positively signed when at least one of its end vertices is a unit of the ring, that is, either one or both of the end vertices can be units for an edge to be positive.…”

“…Conversely, an edge in the unitary Cayley ring signed graph is positively signed only when exactly one of its end vertices is a unit of the ring. Therefore, the difference and the relation between the unitary join Cayley signed graph, the unitary ring Cayley signed graph, and the unitary Cayley meet signed graph were given in [145], and the conditions under which they are isomorphic were obtained, as given in Theorems 49 and 50.…”

“…Along with the above-mentioned characterisations of balanced and canonically consistent unitary Cayley ring signed graphs, the characterisations of clusterable and signcompatible unitary Cayley ring signed graphs were also obtained in [145], as given in Theorems 51 and 52, based on the results on the property of balance. A signed graph is said to be clusterable if its vertex set can be partitioned into pairwise disjoint subsets, called clusters, such that every negative edge joins vertices in different clusters and every positive edge joins vertices in the same cluster.…”

“…Along with the significant characterisations on the properties of balance, clusterability, etc. of the four different signed graphs defined from the unitary Cayley graphs, a few cursory studies on certain derived signed graphs from the signed graphs corresponding to each of the definitions of the signed graphs were also carried out in [143,[145][146][147], which included the discussions on different variations of the line signed graphs, as the canonical marking served as the signs of the edges in the line signed graphs. Moreover, the property of canonical consistency of the signed graph can be used to investigate the properties, such as balance, clusterability, etc., of the line signed graphs.…”