Exercise $10\mathrm{.}2\mathrm{.}2$: Suppose $G$ is a graph of four nodes: $1,2,3,$ and $4.$ Let

$x_{ij},$ for $ij{x}_{14}GGGG1ibe$ 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. $10\mathrm{.}1$

$is$ 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:

$a$