The edge Szeged index of a connected graph G is defined as the sum of products mu(e|G)m_v(e|G) over all edges e = uv of G, where m _u(e|G) is the number of edges whose distance to vertex u is smaller than the distance to vertex v, and m_v(e|G) is the number of edges whose distance to vertex v is smaller than the distance to vertex u. In this paper, we determine the n-vertex unicyclic graphs with the largest, the second largest, the smallest and the second smallest edge Szeged indices.
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