abstract. in this paper we study some relations between the power andquotient power graph of a finite group. these interesting relations motivateus to find some graph theoretical properties of the quotient power graphand the proper quotient power graph of a finite group g. in addition, weclassify those groups whose quotient (proper quotient) power graphs areisomorphic to trees or paths.

for an arbitrary ring $r$, the zero-divisor graph of $r$, denoted by $gamma (r)$, is an undirected simple graph that its vertices are all nonzero zero-divisors of $r$ in which any two vertices $x$ and $y$ are adjacent if and only if either $xy=0$ or $yx=0$. it is well-known that for any commutative ring $r$, $gamma (r) cong gamma (t(r))$ where $t(r)$ is the (total) quotient ring of $r$. in this...

In smart power grids, keeping the synchronicity of generators and the corresponding controls is of great importance. To do so, a simple model is employed in terms of swing equation to represent the interactions among dynamics of generators and feedback control. In case of having a communication network available, the control can be done based on the transmitted measurements by the communication...

We prove the existence of a four-class association scheme on the set of external lines with respect to a hyperbolic quadric of PGð3; qÞ where qd 4 is a power of 2. This result is an analogue of the one by Ebert, Egner, Hollmann and Xiang. Taking a quotient of this association scheme yields a strongly regular graph of Latin square type. We show that this strongly regular graph can also be obtain...

In this paper we survey the behavior of various graph operations on the graph parameters clique-width and NLC-width. We give upper and lower bounds for the clique-width and NLC-width of the modi ed graphs in terms of the clique-width and NLC-width of the involved graphs. Therefor we consider the binary graph operations join, co-join, sum, di erence, products, corona, substitution, and 1-sum, an...

An immediate generalization of the classical McKay correspondence for Gorenstein quotient spaces C/G in dimensions r ≥ 4 would primarily demand the existence of projective, crepant, full desingularizations. Since this is not always possible, it is natural to ask about special classes of such quotient spaces which would satisfy the above property. In this paper we give explicit necessary and suf...

Using the intrinsic relationship between the external equitable partition (EEP) and the spectral properties of the graph Laplacian, we characterize convergence and observability properties of consensus dynamics on networks. In particular, we establish the relationship between the original consensus dynamics and the associated consensus of the quotient graph under varied initial conditions, and ...

An extension of a group A by a group G is thought of here simply as a group H containing A as a normal subgroup with quotient H/A isomorphic to G. It is called a central Frattini extension if A is contained in the intersection of the centre and the Frattini subgroup of H . The result of the paper is that, given a finite abelian A and finite G, there exists a central Frattini extension of A by G...

The object of this paper is the relation between the Szegö kernel of an ample line bundle on a complex projective manifold, M , and the Szegö kernel of the induced polarization on the quotient of M by the holomorphic action of a compact Lie group, G. Let M be an n-dimensional complex projective manifold and L an ample line bundle on it. Suppose a connected compact Lie group G acts on M as a gro...