Elsevier

Applied Mathematics and Computation

Volume 354, 1 August 2019, Pages 329-337
Applied Mathematics and Computation

The weighted vertex PI index of (n,m)-graphs with given diameter

https://doi.org/10.1016/j.amc.2019.02.044Get rights and content

Abstract

The weighted vertex PI index of a graph G is defined byPIw(G)=e=uvE(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.

Section snippets

Introduction and background

Let G=(V,E) be a simple connected graph with vertex set V(G) and edge set E(G). For vertices u, v ∈ V, the distance d(u, v) is defined as the length of a shortest path between u and v in G. Other notations and terminologies used in the paper can be found in [1].

A topological index is a real number related to a graph. It must be a structural invariant, i.e., it preserves by every graph automorphism. Several topological indices have been defined and many of them have found applications as means

The upper bound of connected (n, m)-graphs with given diameter

The following lemma, which is first noticed by [20], is important to the main results of the paper.

Lemma 1

[20] Suppose that G is a connected graph and T an induced subgraph of G such that T is a tree and T is connected to the rest of G only by a cut vertex v. If T is replaced by a star of the same order, centered at v, then the weighted vertex PI index of G increases (unless T is already such a star). If T is replaced by a path of the same order, with one end at v, then the weighted vertex PI index of

The lower bound of trees with given diameter

In this section, two procedures are given which can give the lower bound on the first Zagreb index and the weighted vertex PI index of the trees in T(n,d), and one of the corresponding extremal graphs. Then we study one special set of the extremal graphs generated by the procedures.

Let GT(n,d) be a tree which attains the lower bound on the first Zagreb index. Via the definition of the first Zagreb index, the maximum degree Δ of G should be as small as possible and the number of vertices which

Concluding remark

In this paper, we given the upper bound and the corresponding extremal graphs on the weighted vertex PI index among the graphs in G(n,m,d) where n1m2nd2. We also give the lower bound and the corresponding extremal graphs on the weighted vertex PI index among the trees with n vertices and diameter d via procedures.

A natural question can be proposed in the following.

Question 1

Let Gn,m,d be the set of connected graphs with n vertices, m edges and diameter d. Determine the lower bound and the

Acknowledgments

This work is supported by a research grant NSFC (11461054) of China.

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