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If the Laplacian matrix of a graph has a full set of orthogonal eigenvectors with entries $$\pm1$$, then the matrix formed by taking the columns as the eigenvectors is a Hadamard matrix and the graph is said to be Hadamard diagonalizable. In this article, we prove that if $n=8k+4$ the only possible Hadamard diagonalizable graphs are $$K_n$$, $$K_{n/2,n/2}$$, $$2K_{n/2}$$, and $$nK_1$$, and we develop a computational method for determining all graphs diagonalized by a given Hadamard matrix of any order. Using these two tools, we determine and present all Hadamard diagonalizable graphs up to order 36. Note that it is not even known how many Hadamard matrices there are of order 36. 

Water influences catalytic reactions in multiple ways, including energetic and mechanistic effects. While simulations have provided significant insight into the roles that H 2 O molecules play in aqueous-phase heterogeneous catalysis, questions still remain as to the extent to which H 2 O structures influence catalytic mechanisms. Specifically, influences of the configurational variability in the water structures at the catalyst interface are yet to be understood. Configurational variability is challenging to capture, as it requires multiscale approaches. Herein, we apply a multiscale sampling approach to calculate reaction thermodynamics and kinetics for COH* dehydrogenation to CO* and CH 3 OH* dehydrogenation to CH 2 OH* on Pt(111) catalysts under liquid H 2 O. We explore various pathways for these dehydrogenation reactions that could influence the overall mechanism of methanol decomposition by including participation of H 2 O structures both energetically and mechanistically. We find that the liquid H 2 O environment significantly influences the mechanism of COH* dehydrogenation to CO* but leaves the mechanism of CH 3 OH* dehydrogenation to CH 2 OH* largely unaltered.