We offer a novel graphical method for determining the number of essential sites in enzymes that contain multiple binding sites for a ligand. This method is applicable both to monomeric enzymes containing multiple "unspecific" sites (for protons, metal ions, etc.) and to oligomeric enzymes containing multiple "specific" sites (for substrates and their cognate analogues). The overall procedure is based on some of the intrinsic properties of the general rate equation for enzyme-catalyzed reactions involving multiple binding sites for ligands as elaborated by W. G. Bardsley and R. E. Childs (Biochem. J., 149, 313-328, 1975). The experimental protocol involves measurement of initial rates of enzyme-catalyzed reactions either at varying concentrations of the substrate or at a fixed concentration of the substrate and varying concentrations of effectors (activator or inhibitor). The data are analyzed by plotting 1/v[L]p versus [L] (for inhibition) and [L]p/v versus 1/[L] (for activation) for different (integral) increments of p (such as 0, 1, 2, 3, ... etc.). According to the analytical procedure developed herein, the magnitude of p that yields a horizontal asymptote on these plots serves as the quantitative measure of the number of essential sites in enzyme molecules. By employing this procedure, we have been able to quantitatively ascertain the number of essential sites required for the activation or inhibition of a variety of monomeric and oligomeric enzymes. Among monomeric enzymes we have established that: (i) of the three binding sites for linoleic acid in the lipoxygenase molecule, one site is essential for catalysis, and the other two sites are inhibitory; (ii) of several plausible protonation sites in the alpha-chymotrypsin molecule, only one protonation site is required for the activation of the enzyme; and (iii) there are two inhibitory sites for guanidine-HCl per ribonuclease A molecule; the enzyme is fully inhibited upon binding of guanidine-HCl at any of these two sites. Among oligomeric enzymes, we have discerned that: (i) the individual subunits of LDH and phosphorylase b are catalytically active, and (ii) the catalytic/functional unit of the creatine kinase molecule is likely to be the dimeric subunit. The theoretical details leading to the graphical analysis and its usage in delineating the functional stoichiometry of enzyme-ligand complexes are discussed.