Related papers: Maximum number of symmetric extensions in the rand…

We introduce $k$-robust clique complexes, a family of simplicial complexes that generalizes the traditional clique complex. Here, a subset of vertices forms a simplex provided it does not contain an independent set of size $k$. We…

We find the maximum number of maximal independent sets in two families of graphs: all graphs with $n$ vertices and at most $r$ cycles, and all such graphs that are also connected. In addition, we characterize the extremal graphs.

We determine the maximum number of maximal independent sets of arbitrary graphs in terms of their covering numbers and we completely characterize the extremal graphs. As an application, we give a similar result for K\"onig-Egerv\'ary graphs…

We derive a quadratic-time algorithm for the genus distribution of any 3-regular, biconnected series-parallel graph, which we extend to any biconnected series-parallel graph of maximum degree at most 3. Since the biconnected components of…

For each $\Delta>0$, we prove that there exists some $C=C(\Delta)$ for which the binomial random graph $G(n,C\log n/n)$ almost surely contains a copy of every tree with $n$ vertices and maximum degree at most $\Delta$. In doing so, we…

The $k$-independence number of a graph $G$ is the maximum size of a set of vertices at pairwise distance greater than $k$. In this paper, for each positive integer $k$, we prove sharp upper bounds for the $k$-independence number in an…

Consider a random uniform sample of $n$ points in a compact region $A$ of Euclidean $d$-space, $d \geq 2$, with a smooth or (when $d=2$) polygonal boundary. Fix $k \in {\bf N}$. Let $T_{n,k}$ be the threshold $r$ at which the geometric…

The k-limited packing number, $L_k(G)$, of a graph $G$, introduced by Gallant, Gunther, Hartnell, and Rall, is the maximum cardinality of a set $X$ of vertices of $G$ such that every vertex of $G$ has at most $k$ elements of $X$ in its…

In this paper we study the spectrum of the random geometric graph $G(n,r)$, in a regime where the graph is dense and highly connected. In the \erdren $G(n,p)$ random graph it is well known that upon connectivity the spectrum of the…

We generalize the notion of quasirandom which concerns a class of equivalent properties that random graphs satisfy. We show that the convergence of a graph sequence under the spectral distance is equivalent to the convergence using the…

This survey concerns regular graphs that are extremal with respect to the number of independent sets, and more generally, graph homomorphisms. More precisely, in the family of of $d$-regular graphs, which graph $G$ maximizes/minimizes the…

Let $G$ be a finite connected graph and let $G^{[\star N,k]}$ be the distance $k$-graph of the $N$-fold star power of $G$. For a fixed $k\geq1$, we show that the large $N$ limit of the spectral distribution of $G^{[\star N,k]}$ converges to…

The article deals with two classes of growing random graphs following the preferential attachment rule with a linear weight function, L-graphs, and hybrid Pennock graphs. We determine the exact final vertex degree distribution and the exact…

Let $n,k,s$ be three integers and $\beta$ be a sufficiently small positive number such that $k\geq 3$, $0<1/n\ll \beta\ll 1/k$ and $ks+k\leq n\leq (1+\beta)ks+k-2$. A $k$-graph is called non-trivial if it has no isolated vertex. In this…

In Hazra, den Hollander and Parvaneh (2025) we analysed the friendship paradox for sparse random graphs. For four classes of random graphs we characterised the empirical distribution of the friendship biases between vertices and their…

The clustering coefficient of a vertex in a graph is the proportion of neighbours of the vertex that are adjacent. The minimum clustering coefficient of a graph is the smallest clustering coefficient taken over all vertices. A complete…

We study the probability distribution of the area and the number of vertices of random polygons in a convex set $K\subset\mathbb{R}^2$. The novel aspect of our approach is that it yields uniform estimates for all convex sets…

A $k$-dispersed labelling of a graph $G$ on $n$ vertices is a labelling of the vertices of $G$ by the integers $1, \dots , n$ such that $d(i,i+1) \geq k$ for $1 \leq i \leq n-1$. $DL(G)$ denotes the maximum value of $k$ such that $G$ has a…

We present some product representations for random variables with the Linnik, Mittag-Leffler and Weibull distributions and establish the relationship between the mixing distributions in these representations. Based on these representations,…

A random geometric digraph $G_n$ is constructed by taking $\{X_1,X_2,... X_n\}$ in $\mathbb{R}^2$ independently at random with a common bounded density function. Each vertex $X_i$ is assigned at random a sector $S_i$ of central angle…