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It is shown that the basis graphs of every family of circulants are characterized by their matching polynomials. Explicit formulas are also given for their matching polynomials. From these results, the analogous formulas for the chromatic polynomials of the complements of the basis graphs, are obtained. It is shown that a basis graph of a family of circulants is chromatically unique if and only if it is connected. Also, some interesting results of a computer investigation are discussed and conjectures are made.