  This study proposes the "adaptive flip graph algorithm", which combines
adaptive searches with the flip graph algorithm for finding fast and efficient
methods for matrix multiplication. The adaptive flip graph algorithm addresses
the inherent limitations of exploration and inefficient search encountered in
the original flip graph algorithm, particularly when dealing with large matrix
multiplication. For the limitation of exploration, the proposed algorithm
adaptively transitions over the flip graph, introducing a flexibility that does
not strictly reduce the number of multiplications. Concerning the issue of
inefficient search in large instances, the proposed algorithm adaptively
constraints the search range instead of relying on a completely random search,
facilitating more effective exploration. Numerical experimental results
demonstrate the effectiveness of the adaptive flip graph algorithm, showing a
reduction in the number of multiplications for a $4\times 5$ matrix multiplied
by a $5\times 5$ matrix from $76$ to $73$, and that from $95$ to $94$ for a $5
\times 5$ matrix multiplied by another $5\times 5$ matrix. These results are
obtained in characteristic two.
