In this paper we propose and study a new structural invariant for graphs, called distance-unbalancedness, as measure of how much graph is (un)balanced in terms distances. Explicit formulas are presented several classes well-known graphs. Distance-unbalancedness trees also studied. A few conjectures stated some open problems proposed.

Resistance distance was introduced by Klein and Randić. The Kirchhoff index Kf(G) of a graph G is the sum of resistance distances between all pairs of vertices. A graph G is called a cactus if each block of G is either an edge or a cycle. Denote by Cat(n; t) the set of connected cacti possessing n vertices and t cycles. In this paper, we give the first three smallest Kirchhoff indices among gra...

We investigate the behavior of electric potentials on distance-regular graphs, and extend some results of a prior paper, [12]. Our main result, Theorem 4 below, shows(together with Corollary 3) that if distance is measured by the electric resistance between points then all points are close to being equidistant on a distance-regular graph with large valency. A number of auxiliary results are als...

Given an undirected graph, the resistance distance between two nodes is the resistance one would measure between these two nodes in an electrical network if edges were resistors. Summing these distances over all pairs of nodes yields the so-called Kirchhoff index of the graph, which measures its overall connectivity. In this work, we consider Erdős-Rényi random graphs. Since the graphs are rand...

In machine learning, a popular tool to analyze the structure of graphs is the hitting time and the commute distance (resistance distance). For two vertices u and v, the hitting time Huv is the expected time it takes a random walk to travel from u to v. The commute distance is its symmetrized version Cuv = Huv +Hvu. In our paper we study the behavior of hitting times and commute distances when t...