$!$ Exercise $10\mathrm{.}2\mathrm{.}2$ : Suppose $G$ is a graph of four nodes: $1,2,3,$ and $4$Let $x_{ij},$ for a propositional variable that $we$ interpret $as$ saying "there $isan$ edge between nodes i and $j."$ Any graph $on$ these four nodes can $be$ represented $by$ a truth assignment. For instance, the graph $of$ Fig. $10\mathrm{.}1is$ represented $by$ making $x_{14}$ false and the other five variables true. For any

property $of$ the graph that involves only the existence $or$ nonexistence $of$ edges, $we$ can express that property $as$ a boolean expression that $is$ true $if$ and only $if$ the truth assignment $to$ the variables describes a graph that has the property. Write expressions for the following properties:

$a)$ G has a Hamilton circuit

b$)$ G is connected

c$)$ G containsa clique of size $3,$ that is$,$ a set of $3$ nodes such that there is an edge between every $2$ of them $($i$.$e$.$ triangle in the graph$)$

d$)$ G contains at least one isolated node, that is$,$ a node with no edges