Related papers: Fundamental Groups of Random Clique Complexes

This thesis addresses the question of the maximal number of $d$-simplices for a simplicial complex which is embeddable into $\mathbb{R}^r$ for some $d \leq r \leq 2d$. A lower bound of $f_d(C_{r + 1}(n)) =…

Rank 1 inhomogeneous random graphs are a natural generalization of Erd\H{o}s R\'enyi random graphs. In this generalization each node is given a weight. Then the probability that an edge is present depends on the product of the weights of…

For $r\geq 1$, the $r$-independence complex of a graph $G$ is a simplicial complex whose faces are subset $I \subseteq V(G)$ such that each component of the induced subgraph $G[I]$ has at most $r$ vertices. In this article, we determine the…

Let integer $n \ge 3$ and integer $r = r(n) \ge 3$. Define the binomial random $r$-uniform hypergraph $H_r(n, p)$ to be the $r$-uniform graph on the vertex set $[n]$ such that each $r$-set is an edge independently with probability $p$. A…

We study structural and enumerative aspects of pure simplicial complexes and clique complexes. We prove a necessary and sufficient condition for any simplicial complex to be a clique complex that depends only on the list of facets. We also…

We study adjacency and Laplacian matrices of Erd\H{o}s-R\'{e}nyi $r$-uniform hypergraphs on $n$ vertices with hyperedge inclusion probability $p$, in the setting where $r$ can vary with $n$ such that $r / n \to c \in [0, 1)$. Adjacency…

A \v{C}ech complex of a finite simple graph $G$ is a nerve complex of balls in the graph, with one ball centered at each vertex. More precisely, let the \v{C}ech complex $\mathcal{N}(G,r)$ be the nerve of all closed balls of radius…

We consider a generalised model of a random simplicial complex, which arises from a random hypergraph. Our model is generated by taking the downward-closure of a non-uniform binomial random hypergraph, in which for each $k$, each set of…

The k-Clique problem is a fundamental combinatorial problem that plays a prominent role in classical as well as in parameterized complexity theory. It is among the most well-known NP-complete and W[1]-complete problems. Moreover, its…

We investigate the number of maximal cliques, i.e., cliques that are not contained in any larger clique, in three network models: Erd\H{o}s-R\'enyi random graphs, inhomogeneous random graphs (also called Chung-Lu graphs), and geometric…

For positive integers k,n, we investigate the simplicial complex NM_k(n) of all graphs G on vertex set [n] such that every matching in G has size less than k. This complex (along with other associated cell complexes) is found to be homotopy…

A graph is $(H_1,H_2)$-free for a pair of graphs $H_1,H_2$ if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. In 2001, Kr\'al', Kratochv\'{\i}l, Tuza, and Woeginger initiated a study into the complexity of Colouring for…

For connectivity of \emph{random geometric graphs}, where there is no density for underlying distribution of the vertices, we consider $n$ i.i.d. \emph{Cantor} distributed points on $[0,1]$. We show that for this random geometric graph, the…

Random directed graphs $D(n,p)$ undergo a phase transition around the point $p = 1/n$, and the width of the transition window has been known since the works of Luczak and Seierstad. They have established that as $n \to \infty$ when $p = (1…

We examine properties of random numerical semigroups under a probabilistic model inspired by the Erdos-Renyi model for random graphs. We provide a threshold function for cofiniteness, and bound the expected embedding dimension, genus, and…

Random walks on regular bounded degree expander graphs have numerous applications. A key property of these walks is that they converge rapidly to the uniform distribution on the vertices. The recent study of expansion of high dimensional…

We introduce a natural class of models of random chain complexes of real vector spaces that some classical ensembles of random matrices, the length $1$ case. We are interested here in the homological properties of these random complexes.…

It is proved that fundamental groups of boolean representable simplicial complexes are free and the rank is determined by the number and nature of the connected components of their graph of flats for dimension $\geq 2$. In the case of…

The exponential family of random graphs is one of the most promising class of network models. Dependence between the random edges is defined through certain finite subgraphs, analogous to the use of potential energy to provide dependence…

We prove that every $n$-vertex graph with at least $\binom{n}{2} - (n - 4)$ edges has a fractional triangle decomposition, for $n \ge 7$. This is a key ingredient in our proof, given in a companion paper, that every $n$-vertex $2$-coloured…