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In the context of a random walk on an undirected graph, Kemeny's constant canmeasure the average travel time for a random walk between two randomly chosenvertices. We are interested in graphs that behave counter-intuitively in regardto Kemeny's constant: in particular, we examine graphs with a cut-vertex atwhich at least two branches are paths, regarding whether the insertion of aparticular edge into a graph results in an increase of Kemeny's constant. Weprovide several tools for identifying such an edge in a family of graphs andfor analising asymptotic behaviour of the family regarding the tendency to havethat edge; and classes of particular graphs are given as examples. Furthermore,asymptotic behaviours of families of trees are described.

Families of graphs with twin pendent paths and the Braess edge