Lets take graph with 3 vertices $v _1, v_2, v_3$. now assign different degree to each vertex for $d( v _1)=0$ , $d(v _2)=1$ ,$d(v_3)= 2$, now use handshaking theorem that is $2*edge=$ $sum$ $of$ $the$ $degree$ $of$ $each$ $ vertex$ , which also conclude that sum of degrees of vertices should be even
now , add each degree of each vertices that is $ \Rightarrow 0+1+2 =3$ (which is not even) to make this even u have to make $v _3=1$ , or $v _1=1,v _2=1,v _3=2$ or $v _1=0,v _2=0,v _3=0$ and so many possibilities
now if some people argue for $v _1=2,v _2=3,v _3=5$ , for that we cant tale such example for 3 vertices graph because simple graph(no loop and parallel edges) is given.