A finite automata recognizes 

Which is the application of NFAs

if P = NP then a polynomial-time algorithm exists that takes an undirected graph as input and finds the largest clique contained in that graph

The problem of determining whether there exists a cycle in an undirected graph is in NP.

The class P is NOT closed under the start operation 

The problem defined as 

HALF-CLIQUE = {⟨G⟩| G is an undirected graph having a complete subgraph with at least m/2 nodes, where m is the number of nodes in G} 

is NP-Complete

The complement of the HAMPATH problem is polynomially verifiable

If a problem A is NP-Complete, there exists a non-deterministic polynomial-time algorithm to solve A.

Which of the following problems is an NP-complete problem 

If A is NP-complete and A \in P , then P = NP.