On the Kirchhoff and the Wiener Indices of Graphs and Block Decomposition

Ashkan Nikseresht,Zahra Sepasdar

doi:10.37236/3508

Abstract

In this article we state a relation between the Kirchhoff and Wiener indices of a simple connected graph $G$ and the Kirchhoff and Wiener indices of those subgraphs of $G$ which are induced by its blocks. Then as an application, we define a composition of a rooted tree $T$ and a graph $G$ and calculate its Kirchhoff index in terms of parameters of $T$ and $G$. Finally, we present an algorithm for computing the resistance distances and the Kirchhoff index and a similar one for computing the weighted distances and the Wiener index of a graph. These algorithms are asymptotically faster than the previously known algorithms, on graphs in which the order of the subgraphs induced by blocks is small with respect to the order of the graph.

Highlights

In this article, all graphs are simple, we assume that G is a connected graph and by V(G), n(G), E(G) and N(x) we mean the set of vertices of G, number of its vertices, the set of edges of G and the set of neighbors of a vertex x, respectively
In this article we state a relation between the Kirchhoff and Wiener indices of a simple connected graph G and the Kirchhoff and Wiener indices of those subgraphs of G which are induced by its blocks
We present an algorithm for computing the resistance distances and the Kirchhoff index and a similar one for computing the weighted distances and the Wiener index of a graph

Summary

Introduction

All graphs are simple, we assume that G is a connected graph and by V(G), n(G), E(G) and N(x) we mean the set of vertices of G, number of its vertices, the set of edges of G and the set of neighbors of a vertex x, respectively. The concept of resistance distance was first introduced by Klein and Randic [7]. The electronic journal of combinatorics 21(1) (2014), #P1.25 ambiguity, is defined to be the effective resistance between vertices a and b as computed by Ohm’s and Kirchhoff’s laws. To describe it more concretely, let I(x, y) = Ixy be a positive real valued function defined on all pairs of adjacent vertices of G and P (x) be a real valued function defined on V(G).

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