Abstract
In this research, we present recent results obtained by the authors. They related to spectral invariants of graphs admitting an arbitrary large cyclic group action. To illustrate them we use the family of circulant graphs = 𝐶𝑛(𝑠1, 𝑠2, . . . , ).The Chebyshev polynomials provide a significant analytical tools for studying theproperties of such graphs and their characteristic polynomials. In particular, this gives a way to find analytical expressions for the number of spanning trees (𝑛), the number of rooted spanning forests 𝑓𝐺(𝑛) and the Kirchhoff index (𝐺𝑛) of a graph. We are interested in the behaviour of these invariants for sufficiently large 𝑛. We provideasymptotic formulas of the above mentioned invariants. These results were motivated by problems arising in theoretical physics, biology and chemistry.
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