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 4, N(C-4; G). In particular we show that 3e(G) - 17n(G) + 35 alpha(G) + N(C-4; G) >= 0 for all triangle-free graphs G. We also characterise the graphs that satisfy this inequality with equality. As a consequence we improve the previously best known lower bounds on the independence ratio i(G) = alpha(G)/n(G) for graphs of average degree at most 4 and girth at least 5, 6 or 7.