Abstract:

The critical group of a graph is defined as the torsion subgroup of the co-kernel of the Laplacian matrix of the graph. In this paper, we investigated the critical groups of two classes of unitary circulant graphs, which are Cayley graphs on the group of integers modulo

*n*, with connecting set being the set of units modulo

*n*. The explicit group structure of such graphs when

*n* is product of two distinct primes and when

*n* is a prime power, are computed using Ramanujan Sums. Furthermore, we investigated the critical groups of circulant graphs with fixed connecting sets, and expressed one of the components of the group as the greatest common divisor of real and imaginary parts of Chebyshev Polynomials.

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