  We prove new lower and upper bounds on the higher gonalities of finite
graphs. These bounds are generalizations of known upper and lower bounds for
first gonality to higher gonalities, including upper bounds on gonality
involving independence number, and lower bounds on gonality by scramble number.
We apply our bounds to study the computational complexity of computing higher
gonalities, proving that it is NP-hard to compute the second gonality of a
graph when restricting to multiplicity-free divisors.
