Families of graphs with twin pendent paths and the Braess edge

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