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Network representations are useful for describing the structure of a large variety of complex systems. Although most studies of real-world networks suppose that nodes are connected by only a single type of edge, most natural and engineered…
Graphs are widespread data structures used to model a wide variety of problems. The sheer amount of data to be processed has prompted the creation of a myriad of systems that help us cope with massive scale graphs. The pressure to deliver…
In this article we introduce a definition of k-uniform thresholds hypergraphs through a binary sequence, a natural extension of the classical definition for thresholds graphs. We characterize some of its eigenvalues and multiplicities by…
Graphs are ubiquitous in nature and can therefore serve as models for many practical but also theoretical problems. For this purpose, they can be defined as many different types which suitably reflect the individual contexts of the…
For polynomials and rational maps of fixed degree over a finite field, we bound both the average number of connected components of their functional graphs as well as the average number of periodic points of their associated dynamical…
Many natural and social systems develop complex networks, that are usually modelled as random graphs. The eigenvalue spectrum of these graphs provides information about their structural properties. While the semi-circle law is known to…
We consider graphs with vertices of degree 1 or 2 and prove that the numbers of components of sizes 2 to q have a limit normal distribution for any q > 1. The result is also extended to multigraphs.
Based on a large dataset containing thousands of real-world networks ranging from genetic, protein interaction, and metabolic networks to brain, language, ecology, and social networks we search for defining structural measures of the…
Theoretical analyses for graph learning methods often assume a complete observation of the input graph. Such an assumption might not be useful for handling any-size graphs due to the scalability issues in practice. In this work, we develop…
An upper dominating set in a graph is a minimal (with respect to set inclusion) dominating set of maximum cardinality. The problem of finding an upper dominating set is generally NP-hard. We study the complexity of this problem in classes…
This paper generalizes and unifies the existing spectral bounds on the $k$-independence number of a graph, which is the maximum size of a set of vertices at pairwise distance greater than $k$. The previous bounds known in the literature…
In this paper we introduce a general framework for the study of limits of relational structures in general and graphs in particular, which is based on a combination of model theory and (functional) analysis. We show how the various…
Enlighted by the recent work of Hao Huang on sensitivity conjecture [arXiv:1907.00847], we propose a new definition of the dimension of graphs and establish a relationship between the chromatic number and the dimension.
The metric dimension of a graph $G$ is the size of a smallest subset $L \subseteq V(G)$ such that for any $x,y \in V(G)$ with $x\not= y$ there is a $z \in L$ such that the graph distance between $x$ and $z$ differs from the graph distance…
The metric dimension of a graph is the least number of vertices in a set with the property that the list of distances from any vertex to those in the set uniquely identifies that vertex. Bailey and Meagher obtained an upper bound on the…
Let X \subset R be a bounded set; we introduce a formula that calculates the upper graph box dimension of X (i.e.the supremum of the upper box dimension of the graph over all uniformly continuous functions defined on X). We demonstrate the…
We investigate how the metric dimension of infinite graphs change when we add edges to the graph. Our two main results: (1) there exists a growing sequence of graphs (under the subgraph relation, but without adding vertices) for which the…
We consider the spectral structure of indefinite second order boundary-value problems on graphs. A variational formulation for such boundary-value problems on graphs is given and we obtain both full and half-range completeness results. This…
We give some new bounds for the clique and independence numbers of a graph in terms of its eigenvalues.
We propose a new family of combinatorial inference problems for graphical models. Unlike classical statistical inference where the main interest is point estimation or parameter testing, combinatorial inference aims at testing the global…