Birecognition of prime graphs, and minimal prime graphs

Given a graph [Formula: see text], a subset [Formula: see text] of [Formula: see text] is a module of [Formula: see text] if for each [Formula: see text], [Formula: see text] is adjacent to all the elements of [Formula: see text] or to none of them. For instance, [Formula: see text], [Formula: see text] and [Formula: see text] ([Formula: see text]) are the trivial modules of [Formula: see text]. A graph [Formula: see text] is prime if [Formula: see text] and all its modules are trivial. Given a prime graph [Formula: see text], consider [Formula: see text] such that [Formula: see text] is prime. Given a graph [Formula: see text] such that [Formula: see text] and [Formula: see text], [Formula: see text] and [Formula: see text] are [Formula: see text]-similar if for each [Formula: see text], [Formula: see text] and [Formula: see text] are both prime or not. The graph [Formula: see text] is said to be [Formula: see text]-birecognizable if every graph, [Formula: see text]-similar to [Formula: see text], is prime. We study the graphs [Formula: see text] that are not [Formula: see text]-birecognizable, where [Formula: see text] such that [Formula: see text] is prime, by using the following notion of a minimal prime graph. Given a prime graph [Formula: see text], consider [Formula: see text] such that [Formula: see text] is prime. Given [Formula: see text], [Formula: see text] is [Formula: see text]-minimal if for each [Formula: see text] such that [Formula: see text], [Formula: see text] is not prime.

Finite groups with prime graphs of diameter 5

In this paper we consider a prime graph of finite groups. In particular, we expect finite groups with prime graphs of maximal diameter.

The Metric Dimension and Local Metric Dimension of Relative Prime Graph

This study aims to determine the value of metric dimensions and local metric dimensions of relative prime graphs formed from modulo integer rings, namely . As a vertex set is and if and are relatively prime. By finding the pattern elements of resolving set and local resolving set, it can be shown the value of the metric dimension and the local metric dimension of graphs are and respectively, where is the number of vertices groups that formed multiple 2,3, … , and is the cardinality of set . This research can be developed by determining the value of the fractional metric dimension, local fractional metric dimension and studying the advanced properties of graphs related to their forming rings.Key Words : metric dimension; modulo ; relative prime graph; resolving set; rings.

On odd prime labeling of graphs

In this paper we give a new variation of the prime labeling. We call a graph \(G\) with vertex set \(V(G)\) has an odd prime labeling if its vertices can be labeled distinctly from the set \(\big\{1, 3, 5, ...,2\big|V(G)\big| -1\big\}\) such that for every edge \(xy\) of \(E(G)\) the labels assigned to the vertices of \(x\) and \(y\) are relatively prime. A graph that admits an odd prime labeling is called an <i>odd prime graph</i>. We give some families of odd prime graphs and give some necessary conditions for a graph to be odd prime. Finally, we conjecture that every prime graph is odd prime graph.