On minimal prime extensions of a four-vertex graph in a prime graph

On minimal prime extensions of a four-vertex graph in a prime graph

Directed prime graph of non-commutative ring

Prime graph of a ring R is a graph whose vertex set is the whole set R any any two elements $x$ and $y$ of $R$ are adjacent in the graph if and only if $xRy = 0$ or $yRx = 0$.  Prime graph of a ring is denoted by $PG(R)$. Directed prime graphs for non-commutative rings and connectivity in the graph are studied in the present paper. The diameter and girth of this graph are also studied in the pa...

ON COMPOSITION FACTORS OF A GROUP WITH THE SAME PRIME GRAPH AS Ln(5)

The prime graph of a finite group $G$ is denoted by$ga(G)$. A nonabelian simple group $G$ is called quasirecognizable by primegraph, if for every finite group $H$, where $ga(H)=ga(G)$, thereexists a nonabelian composition factor of $H$ which is isomorphic to$G$. Until now, it is proved that some finite linear simple groups arequasirecognizable by prime graph, for instance, the linear groups $L_...

The Prime Number Graph

Let pn denote the nth prime. The prime number graph is the set of lattice points (n, pn), n = 1, 2.We show that for every k there are k such points that are collinear. By considering the convex hull of the prime number graph, we show that there are infinitely many n such that 2pn < pn_¡ + Pn+Ifor all positive i < n. By a similar argument, we show that there are infinitely many n for which pn > ...

Simple groups with the same prime graph as $D_n(q)$

Vasil'ev posed Problem 16.26 in [The Kourovka Notebook: Unsolved Problems in Group Theory, 16th ed.,Sobolev Inst. Math., Novosibirsk (2006).] as follows:Does there exist a positive integer $k$ such that there are no $k$ pairwise nonisomorphicnonabelian finite simple groups with the same graphs of primes? Conjecture: $k = 5$.In [Zvezdina, On nonabelian simple groups having the same prime graph a...