18 research outputs found
Number of walks and degree powers in a graph
This note deals with the relationship between the total number of -walks
in a graph, and the sum of the -th powers of its vertex degrees. In
particular, it is shown that the the number of all -walks is upper bounded
by the sum of the -th powers of the degrees
Number of walks and degree powers in a graph
This letter deals with the relationship between the total number of k-walks in a graph, and the sum of the k-th powers of its
vertex degrees. In particular, it is shown that the sum of all k-walks is upper bounded by the sum of the k-th powers of the degrees
Network partition via a bound of the spectral radius
12 pages, 10 figures© The author 2016. Published by Oxford University Press. Based on the density of connections between the nodes of high degree, we introduce two bounds of the spectral radius. We use these bounds to split a network into two sets, one of these sets contains the high degree nodes, we refer to this set as the spectral-core. The degree of the nodes of the subnetwork formed by the spectral-core can give an approximation to the top entries of the leading eigenvector of the network.We also present some numerical examples showing the dependancy of the spectral-core with the assortativity coefficient, its evaluation in several real networks and how the properties of the spectral-core can be used to reduce the spectral radius
On local weak limit and subgraph counts for sparse random graphs
We use an inequality of Sidorenko to show a general relation between local
and global subgraph counts and degree moments for locally weakly convergent
sequences of sparse random graphs. This yields an optimal criterion to check
when the asymptotic behaviour of graph statistics such as the clustering
coefficient and assortativity is determined by the local weak limit. As an
application we obtain new facts for several common models of sparse random
intersection graphs where the local weak limit, as we see here, is a simple
random clique tree corresponding to a certain two-type Galton-Watson branching
process