The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. Let U(k) be the set of all unicyclic graphs with a perfect matching. Let C g(G) be the unique cycle of G with length g(G), and M(G) be a perfect matching of G. Let U 0(k) be the subset of U(k) such that g(G)≡ 0 (mod 4), there are just g/2 independence edges of M(G) in C g(G) and there are some edges of E(G)\ M(G) in G\ C g(G) for any G∈U 0(k). In this paper, we discuss the graphs with minimal and second minimal energies in U *(k) = U(k)\ U 0(k), the graph with minimal energy in U 0(k), and propose a conjecture on the graph with minimal energy in U(k).
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Li, X., Zhang, J. & Zhou, B. On unicyclic conjugated molecules with minimal energies. J Math Chem 42, 729–740 (2007). https://doi.org/10.1007/s10910-006-9116-7
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DOI: https://doi.org/10.1007/s10910-006-9116-7