Professor Amongus has just designed an algorithm that can take as input any graph G with n vertices and a number k and determine in O$($nk$)$ time whether G contains a clique of size k$.$ Does Professor Amongus deserve the Turing Award for having just shown that P $=$ NP $?$ Why or why not $?$

For answering the above question, we need to understand: What is a Clique $?$

A clique, C$,$ in an undirected graph G $=($V$,$ E$)$ is a subset of the vertices, C $$ V$,$ such that every two distinct vertices are adjacent. This is equivalent to the condition that the C is a subgraph of G such that C is a complete graph.

Many versions of Clique problem are NP$-$hard.

k$-$clique version is NP$-$complete.

k$-$clique problem:

Input: Undirected Graph $($V$,$ E$),$ and k

The output is a clique with k vertices, if one exists, or an error value indicating it doesn$$t exist.