Related papers: Extremal problems related to Betti numbers of flag…
Consider the family of all finite graphs with maximum degree $\Delta(G)<d$ and matching number $\nu(G)<m$. In this paper we give a new proof to obtain the exact upper bound for the number of edges in such graphs and also characterize all…
Given a graph $G$, the \textit{independence complex} $I(G)$ is the simplicial complex whose faces are the independent sets of $V(G)$. Let $\tilde{b}_i$ denote the $i$-th reduced Betti number of $I(G)$, and let $b(G)$ denote the sum of…
We prove new lower and upper bounds on the higher gonalities of finite graphs. These bounds are generalizations of known upper and lower bounds for first gonality to higher gonalities, including upper bounds on gonality involving…
This work is devoted to lower bounds on independence numbers of distance graphs with vertices in $\{-1,0,1\}^n$. The asymptotic case is studied, yielding new results over a broad range of parameters. Numerical results are presented,…
In this paper, we establish a couple of results on extremal problems in bipartite graphs. Firstly, we show that every sufficiently large bipartite graph with average degree $D$ and with $n$ vertices on each side has a balanced independent…
We consider the following generalized Tur\'an problem: For $2 \le s \le t$, what is the maximum number of triangles in a $K_{1,s,t}$-free graph on $n$ vertices? The previously best known lower and upper bounds are $\Omega(n^2)$ and…
Determining the size of a maximum independent set of a graph $G$, denoted by $\alpha(G)$, is an NP-hard problem. Therefore, many attempts are made to find upper and lower bounds, or exact values of $\alpha (G)$ for special classes of…
We study a spectral analog of the Tur\'an problem for simplicial complexes. Specifically, we consider the extremal problem of maximizing the signless Laplacian spectral radius among simplicial complexes without holes. We determine the…
The independent set reconfiguration problem asks whether one can transform one given independent set of a graph into another, by changing vertices one by one in such a way the intermediate sets remain independent. Extremal problems on…
Random simplicial complexes, as generalizations of random graphs, have become increasingly popular in the literature in recent years. In this paper, we consider a new model for a random simplicial complex that was introduced in…
In the 70s, Berge introduced 1-extendable graphs (also called B-graphs), which are graphs where every vertex belongs to a maximum independent set. Motivated by an application in the design of wireless networks, we study the computational…
We prove graded bounds on the individual Betti numbers of affine and projective complex varieties. In particular, we give for each $p,d,r$, explicit bounds on the $p$-th Betti numbers of affine and projective subvarieties of $\mathrm{C}^k$,…
We obtain some new upper bounds on the maximum number $f(n)$ of edges in $n$-vertex graphs without containing cycles of length four. This leads to an asymptotically optimal bound on $f(n)$ for a broad range of integers $n$ as well as a…
Counting independent sets in graphs and hypergraphs under a variety of restrictions is a classical question with a long history. It is the subject of the celebrated container method which found numerous spectacular applications over the…
We introduce and study a new combinatorial invariant the theta-number $\theta(X)$ of simplicial complexes, and prove that the inequality $\mathcal{C}(X)\leq \theta(X)$ holds for every simplicial complex $X$, where $\mathcal{C}(X)$ denotes…
We give explicit formulas for the asymptotic Betti numbers of the unordered configuration spaces of an arbitrary finite graph over an arbitrary field.
We introduce a new method for computing bounds on the independence number and fractional chromatic number of classes of graphs with local constraints, and apply this method in various scenarios. We establish a formula that generates a…
We provide a simple algorithm for finding the optimal upper bound for sums of products of matrix entries of the form S_pi(N) := sum_{j_1, ..., j_2m = 1}^N t^1_{j_1 j_2} t^2_{j_3 j_4} ... t^m_{j_2m-1 j_2m} where some of the summation indices…
We study the problem of maximizing the number of independent sets in $n$-vertex $k$-chromatic $\ell$-connected graphs. First we consider maximizing the total number of independent sets in such graphs with $n$ sufficiently large, and for…
We consider the topology of simplicial complexes with vertices the points of a random point process and faces determined by distance relationships between the vertices. In particular, we study the Betti numbers of these complexes as the…