Complexity theory diagram

What is the complexity theory of graph theory?
Complexity theory is interested in how algorithms scale with an increase in the input size.
For instance, in the problem of finding whether a graph is connected, how much more time does it take to solve a problem for a graph with 2n vertices compared to the time taken for a graph with n vertices?.
What is the purpose of complexity?
Complexity is an evolving scientific theory addressing properties and behavioral phenomena of systems not well understood, explained, or predictable by means of conventional analysis of its' parts or subsystems.
Complex systems involve many components that are dynamically interacting across size and time scales..
Complexity Theory seeks to understand what makes certain problems algorithmically difficult to solve.
In Data Structures and Algorithms, we saw how to measure the complexity of specific algorithms, by asymptotic measures of number of steps.
The basic tenets of complexity theory are non-linear dynamics, chaos theory, and adaptation/evolution [15]; others include emergence, self-organization, feedback, and chaos [21].

What is the complexity theory of graph theory?