Abstract The semidirect product of a finitely generated group dual with the symmetric can be described through so-called group-theoretical categories partitions (covers only special case; due to Raum–Weber, 2015) and skew (more general; Maaßen, 2018). We generalize these results case graph categories, which allows replace by automorphisms some graph.

The co-maximal subgroup graph $$\Gamma (G)$$ of a group G is whose vertices are non-trivial proper subgroups and two H K adjacent if $$HK=G$$ . In this paper, we continue the study , especially when has isolated vertices. We define new ^*(G)$$ which obtained by removing from characterize connected, complete graph, star an universal vertex etc. also find various parameters like diameter, girth, ...

Let G be a finite insoluble group with soluble radical R(G). In this paper we investigate the graph of G, which is natural generalisation widely studied commuting graph. Here vertices are elements in G∖R(G), x adjacent to y if they generate subgroup G. Our main result states that always connected and its diameter, denoted δS(G), at most 5. More precisely, show δS(G)⩽3 not almost simple obtain s...

A graph is called symmetric if its full automorphism group acts transitively on the set of arcs. The Cayley graph $Gamma=Cay(G,S)$ on group $G$ is said to be normal symmetric if $N_A(R(G))=R(G)rtimes Aut(G,S)$ acts transitively on the set of arcs of $Gamma$. In this paper, we classify all connected tetravalent normal symmetric Cayley graphs of order $p^2q$ where $p>q$ are prime numbers.