Abstract
Hansen et al. used the computer program AutoGraphiX to study the differences between the Szeged index Sz(G) and the Wiener index W(G), and between the revised Szeged index Sz∗(G) and the Wiener index for a connected graph G. They conjectured that for a connected nonbipartite graph G with n≥5 vertices and girth g≥5, Sz(G)−W(G)≥2n−5, and moreover, the bound is best possible when the graph is composed of a cycle C5 on 5 vertices and a tree T on n−4 vertices sharing a single vertex. They also conjectured that for a connected nonbipartite graph G with n≥4 vertices, Sz∗(G)−W(G)≥n2+4n−64, and moreover, the bound is best possible when the graph is composed of a cycle C3 on 3 vertices and a tree T on n−2 vertices sharing a single vertex. In this paper, we not only give confirmative proofs to these two conjectures but also characterize those graphs that achieve the two lower bounds.
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