The Wiener Index Of Trees And Unicyclic Graphs

Let G is a connected graph. The Wiener index of G is the sum of distances between all pairs of vertices in G. Let (?)(n,i)be the set of all trees with order n and matching number i and let U(n,c) be the set of unicyclic graphs with order n and c cut vertices. In this article, we give some graphic transformations that change the Wiener index of graphs, then from these graphic transformations we determine the second to sixth trees in (?)(2i+1,i) and the third to eighth trees in (?)(n,i) for n≥2i+2 having smaller Wiener indices; finally from these graphic transformations we also characterize all extremal graphs having the smallest Wiener index in U(n,c)...