Properties of Random Difference Graphs

Abstract

Generate a bipartite graph on a partitioned set of vertices by randomly assigning to each vertex $v$ some weight $w(v) \in [0,1]$ and adding an edge between vertices $u$ and $v$ (in distinct parts) if and only if $w(v) + w(v) > 1$; the results of such processes are known as difference graphs.Random difference graphs of a given size can be produced either by uniformly random generation of weights or by choosing a graph uniformly at random from the set of all such graphs. We prove that these two methods give rise to the same distribution, and use this equivalence to find exact results for the likelihood of connectivity and Hamiltonicity. We also find the distribution of other properties, such as matching number and degeneracy.

Random Threshold Digraphs

“…The goal of this paper is to extend the work on random threshold graphs [12] (see also [11]) to directed graphs. (See also [11] and [13] which consider a variant of the random threshold model for bipartite graphs; such graphs are known as random difference graphs. We also refer the reader to [7] for another approach to random threshold graphs in the context of graph limits.…”

Counting Spanning Trees of Threshold Graphs

“…Difference graphs are close cousins of threshold graphs, and many threshold graph theorems have adaptations to difference graphs; see for example [19,30]. The following spanning tree formula for difference graphs is obviously analogous to Theorem 1.…”

“…The vertices associated with character "1", the partite set X, are drawn raised in the figure, vertices associated with character "0", the partite set Y , are drawn lowered in the figure, and for all i < j the vertices associated with c i and c j are adjacent if and only if c i is "0" and c j is "1". Proposition 8 (Ross [30]). A graph H is a difference graph if and only if there is a bipartite creation sequence that can construct it.…”