Minimizing the second Zagreb eccentricity index in bipartite graphs with a fixed size and diameter

For a given graph G, the second Zagreb eccentricity index \(\xi _2 (G)\) is defined as the product of the eccentricities of two adjacent vertex pairs in G. This paper mainly studies the problem of determining the graphs that minimize the second Zagreb eccentricity index among n-vertex bipartite graphs with a fixed number of edges and diameter. To be specific, we determine the sharp lower bound on the second Zagreb eccentricity index over the bipartite graphs of order n in terms of fixed edges and diameter. The extremal graphs attaining these lower bounds are fully characterized.