A sharp asymptotic spectral bound for forbidden subgraphs
For every fixed finite graph F of chromatic number r+1 at least three and every positive epsilon, there is an edge threshold M such that every finite simple F-free graph G with m at least M satisfies F2(A_G) <= 2(1-1/r)m - (2(1-1/r)-1)Delta(G) + epsilon m. F2 is the sum of the squares of the two largest positive adjacency eigenvalues with missing terms padded by zero. The graphs may be disconnected. A companion theorem establishes agreement with the two largest ordered eigenvalue squares in the same eventual range. Enough edges force an F-free graph to be noncomplete; BN nonedge interlacing…
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