Related papers: On the independence number of graphs related to a …

Given a projective plane $\Sigma$ and a polarity $\theta$ of $\Sigma$, the corresponding polarity graph is the graph whose vertices are the points of $\Sigma$, and two distinct points $p_1$ and $p_2$ are adjacent if $p_1$ is incident to…

Given a finite projective plane $\Pi$ and a polarity $\theta$ of $\Pi$, the corresponding polarity graph is the graph whose vertices are the points of $\Pi$. Two distinct vertices $p$ and $p'$ are adjacent if $p$ is incident to $\theta…

We derive bounds on the size of an independent set based on eigenvalues. This generalizes a result due to Delsarte and Hoffman. We use this to obtain new bounds on the independence number of the Erd\H{o}s-R\'{e}nyi graphs. We investigate…

Let $G=(V,E)$ be a finite simple graph. In this paper, we study the degree of the $h$-polynomial of the edge ideal of $G$ in relation to the independence number of $G$. Our approach is based on the value of the independence polynomial of…

For a prime power $q$, let $ER_q$ denote the Erd\H{o}s-R\'enyi orthogonal polarity graph. We prove that if $q$ is an even power of an odd prime, then $\chi ( ER_{q}) \leq 2 \sqrt{q} + O ( \sqrt{q} / \log q)$. This upper bound is best…

We determine the independence number of the orthogonality graph on $2^k$-dimensional hypercubes. This answers a question by Galliard from 2001 which is motivated by a problem in quantum information theory. Our method is a modification of a…

Let $G$ be a simple graph and let $L(G)$ denote the \emph{line graph} of $G$. A \emph{$p$-independent} set in $G$ is a set of vertices $S \subseteq V(G)$ such that the subgraph induced by $S$ has maximum degree at most $p$. The…

The following natural problem was raised independently by Erd\H{o}s-Hajnal and Linial-Rabinovich in the late 80's. How large must the independence number $\alpha(G)$ of a graph $G$ be whose every $m$ vertices contain an independent set of…

Many problems in extremal combinatorics can be reduced to determining the independence number of a specific auxiliary hypergraph. We present two such problems, one from discrete geometry and one from hypergraph Tur\'an theory. Using results…

The independence number $\alpha(H)$ of a hypergraph $H$ is the maximum cardinality of a set of vertices of $H$ that does not contain an edge of $H$. Generalizing Shearer's classical lower bound on the independence number of triangle-free…

Let $m_GI$ denote the number of Laplacian eigenvalues of a graph $G$ in an interval $I$ and let $\alpha(G)$ denote the independence number of $G$. In this paper, we determine the classes of graphs that satisfy the condition…

The combinatorial properties of double vertex graphs has been widely studied since the 90's. However only very few results are know about the independence number of such graphs. In this paper we obtain the independence numbers of the double…

We give a probabilistic construction of a $3$-uniform hypergraph on $N$ vertices with independence number $O(\log N / \log \log N)$ in which there are at most two edges among any four vertices. This bound is tight and solves a longstanding…

We consider the number of independent sets in hypergraphs, which allows us to define the independence density of countable hypergraphs. Hypergraph independence densities include a broad family of densities over graphs and relational…

The {\em independence ratio} of a graph $G$ is defined by \[ \iota(G) := \sup_{X \subset V(G)} \frac{|X|}{\alpha(X)},\] where $\alpha(X)$ is the independence number of the subgraph of $G$ induced by $X$. The independence ratio is a…

We introduce a lower bound for the independence number of an arbitrary $k$-uniform hypergraph that only depends on the number of vertices and number of edges of the hypergraph.

In this paper, we prove that for every connected graph G, there exists a split graph H with the same independence number and the same order. Then we propose a first algorithm for finding this graph, given the degree sequence of the input…

Let $\perp$ be a unitary polarity of a finite projective plane $\pi$ of order $q^2$. The unitary polarity graph is the graph with vertex set the points of $\pi$ where two vertices $x$ and $y$ are adjacent if $x \in y^\perp$. We show that a…

Paul Erd\H{o}s suggested the following problem: Determine or estimate the number of maximal triangle-free graphs on $n$ vertices. Here we show that the number of maximal triangle-free graphs is at most $2^{n^2/8+o(n^2)}$, which matches the…

Testing for independence between graphs is a problem that arises naturally in social network analysis and neuroscience. In this paper, we address independence testing for inhomogeneous Erd\H{o}s-R\'{e}nyi random graphs on the same vertex…