Abstract
In 1972, Gutman and Trinajstic showed that total pi-energy of a molecule depends upon a numeric quantity which is often called as Zagreb index. In the same report, they also discussed another numeric quantity depending on the number of atoms at a distance two from a particular atom and proved influencing results on pi-energy of a molecule. In modern literature, this quantity is named as connection number. In this article, we will describe some Zagreb connection numbers for hourglass section in carbon nanocone network with different lengths of cycle in the central core.