A t-Cayley hypergraph X = t-Cay(G; S) is called normal for a finite group G, if the right regular representationR(G) of G is normal in the full automorphism group Aut(X) of X. In this paper, we investigate the normality of t-Cayley hypergraphs of abelian groups, where S < 4.

Abstract : We call a Cayley graph Γ = Cay (G, S) normal for G, if the right regular representation R(G) of G is normal in the full automorphism group of Aut(Γ). In this paper, a classification of all non-normal Cayley graphs of finite abelian group with valency 6 was presented.  

a longstanding conjecture asserts that every finite nonabelian $p$-group admits a noninner automorphism of order $p$‎. ‎let $g$ be a finite nonabelian $p$-group‎. ‎it is known that if $g$ is regular or of nilpotency class $2$ or the commutator subgroup of $g$ is cyclic‎, ‎or $g/z(g)$ is powerful‎, ‎then $g$ has a noninner automorphism of order $p$ leaving either the center $z(g)$ or the frattin...

a longstanding conjecture asserts that every finite nonabelian $p$-group admits a noninner automorphism of order $p$. let $g$ be a finite nonabelian $p$-group. it is known that if $g$ is regular or of nilpotency class $2$ or the commutator subgroup of $g$ is cyclic, or $g/z(g)$ is powerful, then $g$ has a noninner automorphism of order $p$ leaving either the center $z(g)$ or the frattini subgro...

‎we characterize those groups $g$ and vector spaces $v$ such that $v$ is a faithful irreducible $g$-module and such that each $v$ in $v$ is centralized by a $g$-conjugate of a fixed non-identity element of the fitting subgroup $f(g)$ of $g$‎. ‎we also determine those $v$ and $g$ for which $v$ is a faithful quasi-primitive $g$-module and $f(g)$ has no regular orbit‎. ‎we do use these to show in ...

Given a A:-regular graph G of order n, what is the minimum number v(G) of extra vertices required to embed G in a /<+1-regular graph? Clearly v(G) = 0 precisely when the complement G of G has a 1-factor—in particular, when n ^ 2k is even. Suppose G has no 1-factor: if n, k have opposite parity we show that v(G) = 1, while if n, k have the same parity (which must then be even with n < 2k) we sho...

A graph G is called locally s-regular if the neighbourhood of each vertex of G induces a subgraph of G which is regular of degree s. We study graphs which are locally s-regular and simultaneously regular of degree r.

In a dispersable book embedding, the vertices of given graph G must be ordered along line ℓ, called spine, and edges drawn in different half-planes bounded by pages book, such that: (i) no two same page cross, (ii) induced each is 1-regular (or equivalently, matching). The minimum number needed any embedding referred to as thickness dbt(G) G. Graph if dbt(G)=Δ(G) holds (note that Δ(G)≤dbt(G) al...