The degree sequence of an undirected network with $n$ vertices is a non$-$

negative integer vector $din{Z}_{+}^n$ such that $d_i$ is the degree of vertex $i.$ In

order to make the degree sequence for a given network unique, one often

sorts $d$ so that $d_1{d}_2$cdots$d_n.$ It is obvious that a degree sequence

must satisfy $0d_{i}n-1$ for all $i.$

$($a$)$ Give an example of a non$-$negative integer vector $din{Z}_{+}^4$ that satisfies

$0d_{i}3$ for all $i,$ but is not the degree sequence of any network.

$($b$)$ Give an example of a non$-$negative integer vector $din{Z}_{+}^5($or greater$)$

that is the degree sequence of at least two distinct networks $($in other

words, show that a degree sequence does not uniquely determine a

network$).$