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Correlated transmission electron microscopy and secondary ion mass spectrometry with submicron spatial resolution (NanoSIMS) investigations of the same presolar graphites spherules from the Murchison meteorite were conducted, to link the isotopic anomalies with the mineralogy and chemical composition of the graphite and its internal grains. Refractory carbide grains (especially titanium carbide) are commonly found within the graphite spherules, and most have significant concentrations of Zr, Mo, and Ru in solid solution, elements primarily produced by s-process nucleosynthesis. The effect of chemical fractionation on the Mo/Ti ratio in these carbides is limited, and therefore from this ratio one can infer the degree of s-process enrichment in the gas from which the graphite condensed. The resulting s-process enrichments within carbides are large (~200 times solar on average), showing that most of the carbide-containing graphites formed in the mass outflows of asymptotic giant branch (AGB) stars. NanoSIMS measurements of these graphites also show isotopically light carbon (mostly in the 100 < 12C/13C < 400 range). The enrichment of these presolar graphites in both s-process elements and 12C considerably exceeds that astronomically observed around carbon stars. However, a natural correlation exists between 12C and s-process elements, as both form in the He intershell region of thermally pulsing AGB stars and are dredged up together to the surface. Their observation together suggests that these graphites may have formed in chemically and isotopically inhomogeneous regions around AGB stars, such as high-density knots or jets. As shown in the companion paper, a gas density exceeding that expected for smooth mass outflows is required for graphite of the observed size to condense at all in circumstellar environments, and the spatially inhomogeneous, high-density regions from which they condense may also be incompletely mixed with the surrounding gas. We have greatly expanded the available data set of presolar graphites (N = 847) and characterized them by their morphology (onion type and cauliflower type). This effort has also revealed two new, rare presolar phases (iron carbide and metallic osmium). Due to the peculiar gas composition needed to form these rare presolar grain types, the graphites containing them are more likely to originate in supernova outflows.