For any right-angled Artin group, we show that its outer automorphism group contains either a finite-index nilpotent subgroup or a nonabelian free subgroup. This is a weak Tits alternative theorem. We find a criterion on the defining graph that determines which case holds. We also consider some examples of solvable subgroups, including one that is not virtually nilpotent and is embedded in a no...

Let G be a non-abelian group and let Z(G) be the center of G. Associate a graph ΓG (called noncommuting graph of G) with G as follows: Take G\Z(G) as the vertices of ΓG and join two distinct vertices x and y, whenever xy = yx. We want to explore how the graph theoretical properties of ΓG can effect on the group theoretical properties of G. We conjecture that if G and H are two non-abelian finit...

The power graph of a group is the simple whose vertices are elements and two adjacent whenever one them positive other. In this paper, we characterize finite nilpotent groups graphs have equal vertex connectivity minimum degree.

In this note, we show that among finite nilpotent groups of a given order or odd order, the cyclic group has minimum number edges in its subgroup graph. We also conjecture holds for arbitrary groups.

We associate a graph ΓG to a non locally cyclic group G (called the non-cyclic graph of G) as follows: take G\Cyc(G) as vertex set, where Cyc(G) = {x ∈ G | 〈x, y〉 is cyclic for all y ∈ G}, and join two vertices if they do not generate a cyclic subgroup. We study the properties of this graph and we establish some graph theoretical properties (such as regularity) of this graph in terms of the gro...

Let S be a finite generating set of a torsion-free, nilpotent group G. We show that every automorphism of the Cayley graph Cay(G;S) is affine. (That is, every automorphism of the graph is obtained by composing a group automorphism with multiplication by an element of the group.) More generally, we show that if Cay(G1;S1) and Cay(G2;S2) are connected Cayley graphs of finite valency on two nilpot...

For a nilpotent group $G$, let $\Xi(G)$ be the difference between complement of generating graph $G$ and commuting with vertices corresponding to central elements removed. That is, has vertex set $G \setminus Z(G)$, two adjacent if only they do not commute generate $G$. Additionally, $\Xi^+(G)$ subgraph induced by its non-isolated vertices. We show that an edge, then is connected diameter $2$ o...

This paper deals with the classification of groups G such that power graphs and proper are line graphs. In fact, we classify all finite nilpotent whose Also, categorize (except non-abelian 2-groups) Moreover, investigate when generalized quaternion Besides, derive a condition on order dihedral for which

The inverse eigenvalue problem of a graph studies the real symmetric matrices whose off-diagonal pattern is prescribed by adjacencies graph. strong spectral property (SSP) an important tool for this problem. This note establishes bifurcation lemma, which states that if spectrum can be realized matrix with SSP some graph, then all nearby spectra also same idea lemma works other properties and no...