The Estrada index of unicyclic graphs

The Estrada index of a graph G is defined as , where λ1,λ2,…,λn are the eigenvalues of G. We determine the unique unicyclic graph with maximum Estrada index, the unique bipartite unicyclic graph with maximum Estrada index, and the unique bipartite unicyclic graph of a given bipartition with maximum Estrada index. We also show that only two graphs are possible to be the unicyclic graph(s) with minimum Estrada index.