Abstract
The weighted vertex PI index of a graph G is defined byPIw(G)=∑e=uv∈E(G)(dG(u)+dG(v))(nu(e|G)+nv(e|G))where nu(e|G) denotes the number of vertices in G whose distance to the vertex u is smaller than the distance to the vertex v. In this paper, we give the upper bound and the corresponding extremal graphs on the weighted vertex PI index of (n, m)-graphs with diameter d. The lower bound and the corresponding extremal graphs on the first Zagreb index and the weighted vertex PI index of trees with diameter d are given by two procedures. The extremal graphs, given by the two procedures, are also the extremal graphs which attain the lower bound on the first Zagreb index among all connected graphs with n vertices and diameter d.
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