We bound the mean distance in a connected graph which is not a tree in function of its order n and its girth g. On one hand, we show that mean distance is at most n+1 3 − g(g−4) 12n(n−1) if g is even and at most n+1 3 − g(g−1) 12n(n−1) if g is odd. On the other hand, we prove that mean distance is at least ng 4(n−1) unless G is an odd cycle.

To be able to understand how infectious diseases spread on networks, it is important to understand the network structure itself in the absence of infection. In this text we consider dynamic network models that are inspired by the (static) configuration network. The networks are described by population-level averages such as the fraction of the population with k partners, k = 0, 1, 2, . . . This...

Abstract We study the concept of continuous mean distance a weighted graph. For connected unweighted graphs, can be defined as arithmetic distances between all pairs vertices. This parameter provides natural measure compactness graph, and has been intensively studied, together with several variants, including its version for graphs. The analog (discrete) is points on edges Despite being very ge...

It is well-known that the second smallest eigenvalue 22 of the difference Laplacian matrix of a graph G is related to the expansion properties of G. A more detailed analysis of this relation is given. Upper and lower bounds on the diameter and the mean distance in G in terms of 22 are derived.