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 $G$ is defined as $WW(G)=\frac{1}{2}\sum_{\{u,v\}\subseteq V(G)}\bigg(d(u,v)+\frac{1}{2}(d(u,v))^2\bigg)$, where $V(G)$ is the set of all vertices of $G$ and $d(u,v)$ is the distance between the vertices $u,v\in V(G)$. In this paper we find an exact expression for hyper Wiener index of $TUHC_6[2p,q]$, the zigzag polyhex nanotube. doi:10.1017/S144618110800027

Finite BCI-groups are solvable

‎Let $S$ be a subset of a finite group $G$‎. ‎The bi-Cayley graph ${rm BCay}(G,S)$ of $G$ with respect to $S$ is an undirected graph with vertex set $Gtimes{1,2}$ and edge set ${{(x,1),(sx,2)}mid xin G‎, ‎ sin S}$‎. ‎A bi-Cayley graph ${rm BCay}(G,S)$ is called a BCI-graph if for any bi-Cayley graph ${rm BCay}(G,T)$‎, ‎whenever ${rm BCay}(G,S)cong {rm BCay}(G,T)$ we have $T=gS^alpha$ for some $gin G$ and $alphain {rm Aut}(G)$‎. ‎A group $G$ is called a BCI-group if every bi-Cayley graph of $G$ is a BCI-graph‎. ‎In this paper‎, ‎we prove that every BCI-group is solvable‎

A characterization of L2(81)L_2(81) by nse

Let $pi_e(G)$ be the set of element orders of a finite group $G$‎. ‎Let $nse(G)={m_nmid ninpi_e(G)}$‎, ‎where $m_n$ be the number of elements of order $n$ in $G$‎. ‎In this paper‎, ‎we prove that if $nse(G)=nse(L_2(81))$‎, ‎then $Gcong L_2(81)$‎