Groups whose prime graph on class sizes has a cut vertex

Groups whose Bipartite Divisor Graph for Character Degrees Has Five Vertices

Let $G$ be a finite group and $cd^*(G)$ be the set of nonlinear irreducible character degrees of  $G$. Suppose that $rho(G)$ denotes the set of primes dividing some element of $cd^*(G)$. The bipartite divisor graph for the set of character degrees which is denoted by $B(G)$, is a bipartite graph whose vertices are the disjoint union of $rho(G)$ and $cd^*(G)$, and a vertex $p in rho(G)$ is conne...

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

On Pathos Adjacency Cut Vertex Jump Graph of a Tree

In this paper the concept of pathos adjacency cut vertex jump graph PJC(T ) of a tree T is introduced. We also present a characterization of graphs whose pathos adjacency cut vertex jump graphs are planar, outerplanar, minimally non-outerplanar, Eulerian and Hamiltonian.

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...

Divisibility Criteria for Class Numbers of Imaginary Quadratic Fields Whose Discriminant Has Only Two Prime Factors

and Applied Analysis 3