Geodetic graphs minimize the geodesic subpath number while an upper bound is established for general graphs on n vertices and extremal structures are characterized among cactus graphs.
Yamamoto, Approximately counting paths and cycles in a graph,Discrete Appl
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Counting geodesic paths in graphs
Geodetic graphs minimize the geodesic subpath number while an upper bound is established for general graphs on n vertices and extremal structures are characterized among cactus graphs.