20 research outputs found
Zagreb Indices of the Generalized Hierarchical Product of Graphs
Abstract. In this paper the first and the second Zagreb indices of generalized hierarchical product of graphs, which is generalization of standard hierarchical and Cartesian product of graphs, is computed. As a consequence we compute the Zagreb indices of some chemical graphs
On a Combinatorial Problem in Group Theory
Let n be a positive integer or infinity (denote ∞). We denote by W ∗ (n) the class of groups G such that, for every subset X of G of cardinality n + 1, there exist a positive integer k, and a subset X0 ⊆ X, with 2 ≤ |X0 | ≤ n + 1 and a function f: {0, 1, 2,..., k} − → X0, with f(0) � = f(1) and non-zero integers t0, t1,..., tk such that [x t0 0, xt1 1,..., xt k] = 1, where xi: = f(i), i = 0,..., k, and xj ∈ H whenever k x t j j ∈ H, for some subgroup H �=�x t j j�of G. If the integer k is fixed for every subset X we obtain the class W ∗ k (n). Here we prove that (1) Let G ∈ W ∗ (n), n a positive integer, be a finite group, p> n a prime divisor of the order of G, P a Sylow p-subgroup of G. Then there exists a normal subgroup K of G such that G = P × K. (2) A finitely generated soluble group has the property W ∗ (∞) if and only if it is finite-by-nilpotent. (3) Let G ∈ W ∗ k (∞) be a finitely generated soluble group, then G is finite-by-(nilpotent of k-bounded class)
Hyper Wiener index of zigzag polyhex nanotubes
The hyper Wiener index of a connected graph is defined as , where is the set of all vertices of and is the distance between the vertices . In this paper we find an exact expression for hyper Wiener index of , the zigzag polyhex nanotube.
doi:10.1017/S144618110800027
Finite BCI-groups are solvable
‎Let be a subset of a finite group ‎. ‎The bi-Cayley graph of with respect to is an undirected graph with vertex set and edge set ‎. ‎A bi-Cayley graph is called a BCI-graph if for any bi-Cayley graph ‎, ‎whenever we have for some and ‎. ‎A group is called a BCI-group if every bi-Cayley graph of is a BCI-graph‎. ‎In this paper‎, ‎we prove that every BCI-group is solvable‎
A characterization of by nse
Let be the set of element orders of a finite group ‎. ‎Let ‎, ‎where be the number of elements of order in ‎. ‎In this paper‎, ‎we prove that if ‎, ‎then ‎