The sizes of Markov equivalence classes of directed acyclic graphs play important roles in measuring the uncertainty and complexity in causal learning. A Markov equivalence class can be represented by an essential graph and its undirected subgraphs determine the size of the class. In this paper, we develop a method to derive the formulas for counting the sizes of Markov equivalence classes. We ...

RDF is the data model of choice for Semantic Web applications. RDF graphs are often large and have heterogeneous, complex structure. Graph summaries are compact structures computed from the input graph; they are typically used to simplify users’ experience and to speed up graph processing. We introduce a formal RDF summarization framework, based on graph quotients and RDF node equivalence; our ...

‎In this paper, first we introduce the end of locally finite graphs as an equivalence class of infinite paths in the graph. Then we mention the ends of finitely generated groups using the Cayley graph. It was proved that the number of ends of groups are not depended on the Cayley graph and that the number of ends in the groups is equal to zero, one, two, or infinity. For ...

An equivalence graph is a disjoint union of cliques. For a graph G let eq(G) be the minimum number of equivalence subgraphs of G needed to cover all edges of G. We call eq( G) the equivalence covering number of G. It was shown in [ 81 that computing the equivalence covering number is NP-hard, even when restricted to graphs in which no two triangles have a vertex in common. We show that the equi...

Two types of equivalence relation are used to classify functions between finite groups into classes which preserve combinatorial and algebraic properties important for a wide range of applications. However, it is very difficult to tell when functions equivalent under the coarser (“graph”) equivalence are inequivalent under the finer (“bundle”) equivalence. Here we relate graphs to transversals ...

Tutte has described in the book “Connectivity in graphs” a canonical decomposition of any graph into 3-connected components. In this article we translate (using the language of symbolic combinatorics) Tutte’s decomposition into a general grammar expressing any family G of graphs (with some stability conditions) in terms of the subfamily G3 of graphs in G that are 3-connected (until now, such a ...

We introduce the notion of orbit equivalence of directed graphs, following Matsumoto’s notion of continuous orbit equivalence for topological Markov shifts. We show that two graphs in which every cycle has an exit are orbit equivalent if and only if there is a diagonal-preserving isomorphism between their C∗-algebras. We show that it is necessary to assume that every cycle has an exit for the f...

We show that asymptotic equivalence, in a strong form, holds between two random graph models with slightly differing edge probabilities under substantially weaker conditions than what might naively be expected. One application is a simple proof of a recent result by van den Esker, van der Hofstad and Hooghiemstra on the equivalence between graph distances for some random graph models.