We show that the outer automorphism groups of graph products finitely generated abelian satisfy Tits alternative, are residually finite, their so-called Torelli subgroups generated, and they a dichotomy between being virtually nilpotent containing non-abelian free subgroup is determined by graphical condition on underlying labelled graph. Graph simultaneously generalize right-angled Artin (RAAG...

Let G be a group. The power graph of G is a graph with the vertex set G, having an edge between two elements whenever one is a power of the other. We characterize nilpotent groups whose power graphs have finite independence number. For a bounded exponent group, we prove its power graph is a perfect graph and we determine its clique/chromatic number. Furthermore, it is proved that for every grou...

assume that $(n,l)$, is a pair of finite dimensional nilpotent lie algebras, in which $l$ is non-abelian and $n$ is an ideal in $l$ and also $mathcal{m}(n,l)$ is the schur multiplier of the pair $(n,l)$. motivated by characterization of the pairs $(n,l)$ of finite dimensional nilpotent lie algebras by their schur multipliers (arabyani, et al. 2014) we prove some properties of a pair of nilpoten...

Assume that $(N,L)$, is a pair of finite dimensional nilpotent Lie algebras, in which $L$ is non-abelian and $N$ is an ideal in $L$ and also $mathcal{M}(N,L)$ is the Schur multiplier of the pair $(N,L)$. Motivated by characterization of the pairs $(N,L)$ of finite dimensional nilpotent Lie algebras by their Schur multipliers (Arabyani, et al. 2014) we prove some properties of a pair of nilpoten...

A number of combinatorial problems are treated using properties of abelian nilpotentand idempotent-generated subalgebras of Clifford algebras. For example, the problem of deciding whether or not a graph contains a Hamiltonian cycle is known to be NP-complete. By considering entries of Λ, where Λ is an appropriate nilpotent adjacency matrix, the k-cycles in any finite graph are recovered. Within...

Edgeworth expansions for random walks on covering graphs with groups of polynomial volume growths are obtained under a few natural assumptions. The coefficients appearing in this expansion depends not only geometric features the underlying but also modified harmonic embedding graph into certain nilpotent Lie group. Moreover, we apply rate convergence Trotter's approximation theorem to establish...

It is shown that an algebra Λ can be lifted with nilpotent Jacobson radical r = r(Λ) and has a generalized matrix unit {eii}I with each ēii in the center of Λ̄ = Λ/r iff Λ is isomorphic to a generalized path algebra with weak relations. Representations of the generalized path algebras are given. As a corollary, Λ is a finite algebra with non-zero unity element over perfect field k (e.g. a field ...

In recent years, the RFRS condition has been used to analyze virtual fibering in 3-manifold topology. Agol’s work shows that any 3-manifold with zero Euler characteristic satisfying the RFRS condition on its fundamental group virtually fibers over the circle. In this note we will show that a finitely generated nilpotent group is either virtually abelian or is not virtually RFRS. As a corollary,...

Nilpotent and idempotent adjacency operator methods are applied to the study of random geometric graphs in a discretized, d-dimensional unit cube [0, 1]. Cycles are enumerated, sizes of maximal connected components are computed, and closed formulas are obtained for graph circumference and girth. Expected numbers of k-cycles, expected sizes of maximal components, and expected circumference and g...