An invariant for minimum triangle-free graphs

We study the number of edges, $e(G)$, in triangle-free graphs with a prescribed number of vertices, $n(G)$, independence number, $\alpha(G)$, and number of cycles of length four, $\operatorname{N}(C_4;G)$. We in particular show that $$3e(G) - 17n(G) + 35\alpha(G) + \operatorname{N}(C_4;G) \geq 0$$ for all triangle-free graphs $G$. We also characterise the graphs that satisfy this inequality with equality.