相关论文: Nested hierarchies in planar graphs

Networks are a fundamental tool for understanding and modeling complex systems in physics, biology, neuroscience, engineering, and social science. Many networks are known to exhibit rich, lower-order connectivity patterns that can be…

Traditional graph analysis focuses on nodes and edges, that is, pairwise relationships. Yet many real-world networks, including biological, social, and communication networks, involve higher-order relationships in which multiple nodes…

Complex networks often exhibit community structure, with communities corresponding to denser subgraphs in which nodes are closely linked. When modelling systems where interactions extend beyond node pairs to arbitrary numbers of nodes,…

We explore the concept of separating systems of vertex sets of graphs. A separating system of a set $X$ is a collection of subsets of $X$ such that for any pair of distinct elements in $X$, there exists a set in the separating system that…

In this paper, we relate the seemingly unrelated concepts of treewidth and boxicity. Our main result is that, for any graph G, boxicity(G) <= treewidth(G) + 2. We also show that this upper bound is (almost) tight. Our result leads to…

A graph puzzle ${\rm Puz}(G)$ of a graph $G$ is defined as follows. A configuration of ${\rm Puz}(G)$ is a bijection from the set of vertices of a board graph to the set of vertices of a pebble graph, both graphs being isomorphic to some…

Hypergraphs, encoding structured interactions among any number of system units, have recently proven a successful tool to describe many real-world biological and social networks. Here we propose a framework based on statistical inference to…

Visualization of the adjacency matrix enables us to capture macroscopic features of a network when the matrix elements are aligned properly. Community structure, a network consisting of several densely connected components, is a…

In this work we develop a theory of hierarchical clustering for graphs. Our modeling assumption is that graphs are sampled from a graphon, which is a powerful and general model for generating graphs and analyzing large networks. Graphons…

High triangle density -- the graph property stating that a constant fraction of two-hop paths belong to a triangle -- is a common signature of social networks. This paper studies triangle-dense graphs from a structural perspective. We prove…

There are various results connecting ranks of incidence matrices of graphs and hypergraphs with their combinatorial structure. Here, we consider the generalized incidence matrix $N_2$ (defined by inclusion of pairs in edges) for one natural…

We present a new characterization of $k$-trees based on their reduced clique graphs and $(k+1)$-line graphs, which are block graphs. We explore structural properties of these two classes, showing that the number of clique-trees of a…

Triadic closure, the formation of a connection between two nodes in a network sharing a common neighbor, is considered a fundamental mechanism determining the clustered nature of many real-world topologies. In this work we define a static…

We introduce a taxonomy of interaction types and show that graphs are focal hypergraphs: every graph is canonically a focal hypergraph via its closed neighbourhood structure, and every graph dynamical model is a special case of the general…

We introduce and study the pinnacle sets of a simple graph $G$ with $n$ vertices. Given a bijective vertex labeling $\lambda\,:\,V(G)\rightarrow [n]$, the label $\lambda(v)$ of vertex $v$ is a pinnacle of $(G, \lambda)$ if…

We characterize the connected graphs of given order $n$ and given independence number $\alpha$ that maximize the number of maximum independent sets. For $3\leq \alpha\leq n/2$, there is a unique such graph that arises from the disjoint…

A complex system with many interacting individuals can be represented by a network consisting of nodes and links representing individuals and pairwise interactions, respectively. However, real-world systems grow with time and include many…

Finding densely connected subsets of vertices in an unsupervised setting, called clustering or community detection, is one of the fundamental problems in network science. The edge clustering approach instead detects communities by…

Despite being very successful within the pattern recognition and machine learning community, graph-based methods are often unusable because of the lack of mathematical operations defined in graph domain. Graph embedding, which maps graphs…

For a finite group $G$, the vertices of the prime graph $\Gamma(G)$ are the primes that divide $|G|$, and two vertices $p$ and $q$ are connected by an edge if and only if there is an element of order $pq$ in $G$. Prime graphs of solvable…