the atom-bond connectivity index of graph is a topological index proposed by estrada et al.as abc (g)  uve (g ) (du dv  2) / dudv , where the summation goes over all edges ofg, du and dv are the degrees of the terminal vertices u and v of edge uv. in the present paper,some upper bounds for the second type of atom-bond connectivity index are computed.

let $g$ be a group. the order graph of $g$ is the (undirected)graph $gamma(g)$,those whose vertices are non-trivial subgroups of $g$ and two distinctvertices $h$ and $k$ are adjacent if and only if either$o(h)|o(k)$ or $o(k)|o(h)$. in this paper, we investigate theinterplay between the group-theoretic properties of $g$ and thegraph-theoretic properties of $gamma(g)$. for a finite group$g$, we s...

In this paper, we show the redundancies of multiset topological spaces. It is proved that $(P^star(U),sqsubseteq)$ and $(Ds(varphi(U)),subseteq)$ are isomorphic. It follows that multiset topological spaces are superfluous and unnecessary in the theoretical view point.

An extended star is a tree which has only one vertex with degree larger than two. The -center problem in a graph  asks to find a subset  of the vertices of  of cardinality  such that the maximum weighted distances from  to all vertices is minimized. In this paper we consider the -center problem on the unweighted extended stars, and present some properties to find solution.

The importance of energy crisis and global warming necessitates presenting strategies in order to decrease the amount of emissions as well as fuel consumption in large and complex industries such as refineries and petrochemical industries. Generally, refinery steam networks are regarded as units which consume fuel enormously. In this paper, the steam network of Tehran Oil Refinery is considered...