Related papers: Independent sets in certain classes of (almost) re…
In this paper, we construct intriguing sets in five classes of strongly regular graphs defined on nonisotropic points of finite classical polar spaces, and determine their intersection numbers.
Graph independence (also known as $\epsilon$-independence or $\lambda$-independence) is a mixture of classical independence and free independence corresponding to graph products or groups and operator algebras. Using conjugation by certain…
We consider Catalan-pair graphs, a family of graphs that can be viewed as representing certain interactions between pairs of objects which are enumerated by the Catalan numbers. In this paper we study random Catalan-pair graphs and deduce…
Graph polynomials are polynomials assigned to graphs. Interestingly, they also arise in many areas outside graph theory as well. Many properties of graph polynomials have been widely studied. In this paper, we survey some results on the…
Let $G$ be a group and $L(G)$ be the set of all subgroups of $G$. We introduce a bipartite graph $\mathcal{B}(G)$ on $G$ whose vertex set is the union of two sets $G \times G$ and $L(G)$, and two vertices $(a, b) \in G \times G$ and $H \in…
In this thesis, we study connections between metric and combinatorial graphs from a Dirichlet space point of view.
A categorical formalism for directed graphs is introduced, featuring natural notions of morphisms and subgraphs, and leading to two elementary descriptions of the free-properad monad, first in terms of presheaves on elementary graphs,…
We consider a variety of connections between threshold graphs, shifted complexes, and simplicial complexes naturally formed from a graph. These graphical complexes include the independent set, neighborhood, and dominance complexes. We…
The abundance of interconnected data has fueled the design and implementation of graph generators reproducing real-world linking properties, or gauging the effectiveness of graph algorithms, techniques and applications manipulating these…
We prove that the invariably generating graph of a finite group can have an arbitrarily large number of connected components with at least two vertices.
In this article, I present a conjecture on the number of independent sets on graph covers. I also show that the conjecture implies that the partition function of a binary pairwise attractive model is greater than that of the Bethe…
We show that graphs, networks and other related discrete model systems carry a natural supersymmetric structure, which, apart from its conceptual importance as to possible physical applications, allows to derive a series of spectral…
We organize a table of regular graphs with minimal diameters and minimal mean path lengths, large bisection widths and high degrees of symmetries, obtained by enumerations on supercomputers. These optimal graphs, many of which are newly…
We present the first combinatorial scheme for counting labelled 4-regular planar graphs through a complete recursive decomposition. More precisely, we show that the exponential generating function of labelled 4-regular planar graphs can be…
We study some properties of graphs (or, rather, graph sequences) defined by demanding that the number of subgraphs of a given type, with vertices in subsets of given sizes, approximatively equals the number expected in a random graph. It…
Firstly, for a general graph, we find a recursion formula on the number of Hamiltonian cycles and one on cycles. By this result, we give some new polynomial invariants. Secondly, we give a condition to tell whether a polynomial defined by…
We consider colored compositions where only some parts are allowed different colors, depending on their locations in the composition. The counting sequences are obtained through generating functions. Connections to many other combinatorial…
An independent dominating set of the simple graph $G=(V,E)$ is a vertex subset that is both dominating and independent in $G$. The independent domination polynomial of a graph $G$ is the polynomial $D_i(G,x)=\sum_{A} x^{|A|}$, summed over…
The configuration space of causal sets is vast. It is a critical goal to map out this space. Here, we take a practical step towards this goal. We investigate nine classes of causal sets, most of them not studied before. These include…
A metrized graph is a compact singular 1-manifold endowed with a metric. A given metrized graph can be modelled by a family of weighted combinatorial graphs. If one chooses a sequence of models from this family such that the vertices become…