Let $$G$$ be a complete undirected graph on $$6$$ vertices. If vertices of $$G$$ $$\,\,\,\,$$ are labeled, then the number of distinct cycles of length $$4$$ in $$G$$ is equal to

The degree sequence of a simple graph is the sequence of the degrees of the nodes in the graph in decreasing order. Which of the following sequences can not be the degree sequence of any graph?
$${\rm I}.$$$$\,\,\,\,\,7,6,5,4,4,3,2,1$$
$${\rm I}{\rm I}.$$$$\,\,\,\,\,6,6,6,6,3,3,2,2$$
$${\rm I}{\rm I}{\rm I}.$$$$\,\,\,\,\,7,6,6,4,4,3,2,2$$
$${\rm I}V.$$$$\,\,\,\,\,8,7,7,6,4,2,1,1$$

$$G$$ is a simple undirected graph. Some vertices of $$G$$ are of odd degree. Add a node $$v$$ to $$G$$ and make it adjacent to each odd degree vertex of $$G$$. The resultant graph is sure to be

A binary tree with $$n>1$$ nodes has $${n_1}$$, $${n_2}$$ and $${n_3}$$ nodes of degree one, two and three respectively. The degree of a node is defined as the number of its neighbours.

Starting with the above tree, while there remains a node $$v$$ of degree two in the tree, add an edge between the two neighbours of $$v$$ and then remove $$v$$ from the tree. How many edges will remain at the end of the process?