def add$\_$edge$\_$to$\_$graph$($unique$\_$weights, edge$\_$weight$)$:

if edge$\_$weight not in unique$\_$weights:

unique$\_$weights.add$($edge$\_$weight$)$

return True

return False

def assign$\_$labels$($n$,$ k$)$:

# initialize an empty list to store vertex labels

vertex$\_$labels $=[]$

# initialize an empty list to store edge weights

edge$\_$weights $=[]$

# create a set to store unique edge weights

unique$\_$weights $=$ set$()$

# get the order of sn$3$

order$\_$of$\_$sn$3=(3*$ n$)+1$

# create a list of integers from $0$ to k$-1$

labels $=$ list$($range$($k$))$

# assign labels to vertices

for i in range$($order$\_$of$\_$sn$3)$:

vertex$\_$labels.append$($labels$[$i $\%$ k$])$

# assign weights to edges

for i in range$($order$\_$of$\_$sn$3)$:

# Ensure only adjacent vertices are connected

if i $+1$ order$\_$of$\_$sn$3$:

edge$\_$weight $=$ abs$($vertex$\_$labels$[$i$]-$ vertex$\_$labels$[$i $+1])$

if add$\_$edge$\_$to$\_$graph$($unique$\_$weights, edge$\_$weight$)$:

edge$\_$weights.append$($edge$\_$weight$)$

else:

# Adjust vertex labels if edge weight is not unique

vertex$\_$labels$[$i$]=($vertex$\_$labels$[$i$]+1)\%$ k

vertex$\_$labels$[$i $+1]=($vertex$\_$labels$[$i $+1]+1)\%$ k

edge$\_$weights.append$($edge$\_$weight$)$

# return the vertex labels and edge weights

return vertex$\_$labels, edge$\_$weights

# Take input for n from the user

n $=$ int$($input$("$Enter the value of n: $"))$

k $=((3*$ n$)+1)//2$ # calculate k based on n

vertex$\_$labels, edge$\_$weights $=$ assign$\_$labels$($n$,$ k$)$

print$("$vertex labels:", vertex$\_$labels$)$

print$("$edge weights:", edge$\_$weights$)$

Problem$-1$: Homogenous amalgamated Star : $S_{n,3}$

For $n3,$ homogenous amalgamated

star $S_{n,3}$ admits the edge irregular k$-$

labeling.

Order of $S_{n,3}=3n+1.$

Vertex label is at most $k$ and the edge

weights are distinctive, thus $S_{n,3}$ admits

the edge irregular $k-$labeling.

$k=|$~$\frac{3n+1}{2}$~$|$

Make changes to the above code so that the edge weights should be unique $,$ and the output graph should look like the graph provided from the snapshot.