Variable neighborhood search for extremal graphs: 1 the AutoGraphiX system
Discrete Mathematics
On a conjecture about the Szeged index
European Journal of Combinatorics
Use of the Szeged index and the revised Szeged index for measuring network bipartivity
Discrete Applied Mathematics
Discrete Applied Mathematics
Mathematical and Computer Modelling: An International Journal
Note: Wiener index versus Szeged index in networks
Discrete Applied Mathematics
Bicyclic graphs with maximal revised Szeged index
Discrete Applied Mathematics
Improved bounds on the difference between the Szeged index and the Wiener index of graphs
European Journal of Combinatorics
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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 C"5 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)=n^2+4n-64, and moreover, the bound is best possible when the graph is composed of a cycle C"3 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.