Narrow Results

Let G be a finite group and H be a subgroup of G. We introduce the non-normal graph of H in G, denoted by _H,G, and give some of the graph theoretical properties of _H,G.

Let G be a group. The permutability graph of subgroups of G, denoted by Γ(G), is a graph with all the proper subgroups of G as its vertices and two distinct vertices in Γ(G) are adjacent if and only if the corresponding subgroups permute in G. In this paper, we classify the finite groups whose permutability graphs of subgroups are planar. In addition, we classify the finite groups whose permutability graphs of subgroups are one of outerplanar, path, cycle, unicyclic, claw-free or C₄-free. Also, we investigate the planarity of permutability graphs of subgroups of infinite groups.

Given a group $G$, the enhanced power graph of $G$, denoted by ${{𝒢}}_e(G)$, is the graph with vertex set $G$ and two distinct vertices $x$ and $y$ are edge connected in ${{𝒢}}_e(G)$ if there exists $z\in G$ such that $x=z^m$ and $y=z^n$ for some $m,n\in \mathbb{N}$. Here, we show that the graph ${{𝒢}}_e(G)$ is complete if and only if $G$ is cyclic; and ${{𝒢}}_e(G)$ is Eulerian if and only if $\left|G\right|$ is odd. We characterize all abelian groups and all non-abelian $p$-groups $G$ such that ${{𝒢}}_e(G)$ is dominatable. Besides, we show that there is a one-to-one correspondence between the maximal cliques in ${{𝒢}}_e(G)$ and the maximal cyclic subgroups of $G$.

Let $𝕍$ be a k-dimensional vector space over a finite field $𝔽$ with a basis $\{{\alpha }_1,\dots ,{\alpha }_k\}$. The nonzero component graph of $𝕍$, denoted by $\mathrm{\Gamma }(𝕍)$, is a simple undirected graph with vertex set as nonzero vectors of $𝕍$ such that there is an edge between two distinct vertices $x,y$ if and only if there exists at least one ${\alpha }_i$ along which both $x$ and $y$ have nonzero scalars. In this paper, we find the vertex connectivity and girth of $\mathrm{\Gamma }(𝕍)$. We also characterize all vector spaces $𝕍$ for which $\mathrm{\Gamma }(𝕍)$ has genus either 0 or 1 or 2.