Can anyone explain this? How to get head, tail, vert and next?
You should invest effort in getting the description of this graph representation. Guessing a definition is generally not the right kind of activity in mathematics/computer science.
It seems that the ith elements of TAIL and HEAD arrays give, correspondingly, the tail and head of edge number i. Concerning VERT and NEXT, my guess is that the ith element of VERT is the number of some edge exiting from vertex #i, or 0 if no edge exists from i. The ith element of NEXT gives the number of the next edge exiting from the same vertex as edge #i, or 0 if there are no more edges.