Related papers: On the extreme eigenvalues of regular graphs

Let $c$ be a proper edge colouring of a graph $G=(V,E)$ with integers $1,2,\ldots,k$. Then $k\geq \Delta(G)$, while by Vizing's theorem, no more than $k=\Delta(G)+1$ is necessary for constructing such $c$. On the course of investigating…

We show that if $G$ is a graph on $n$ vertices, with all degrees comparable to some $d = d(n)$, and without a sparse cut, for a suitably chosen notion of sparseness, then it contains a complete minor of order \[ \Omega\left( \sqrt{\frac{n…

For a graph $G=(V,E)$, let $bc(G)$ denote the minimum number of pairwise edge disjoint complete bipartite subgraphs of $G$ so that each edge of $G$ belongs to exactly one of them. It is easy to see that for every graph $G$, $bc(G) \leq n…

A $(k,g,\underline{g+1})$-graph is a $k$-regular graph of girth $g$ which does not contain cycles of length $g+1$. Such graphs are known to exist for all parameter pairs $k \geq 3, g \geq 3 $, and we focus on determining the orders…

The Erd\H{o}s--Gallai Theorem states that for $k\geq 3$ every graph on $n$ vertices with more than $\frac{1}{2}(k-1)(n-1)$ edges contains a cycle of length at least $k$. Kopylov proved a strengthening of this result for 2-connected graphs…

We prove that there is a constant $c >0$, such that whenever $p \ge n^{-c}$, with probability tending to 1 when $n$ goes to infinity, every maximum triangle-free subgraph of the random graph $G_{n,p}$ is bipartite. This answers a question…

In a graph $G$, a subset of vertices $S \subseteq V(G)$ is said to be cyclable if there is a cycle containing the vertices in some order. $G$ is said to be $k$-cyclable if any subset of $k \geq 2$ vertices is cyclable. If any $k$…

For a family of graphs $\mathcal{F}$, a graph $G$ is $\mathcal{F}$-universal if $G$ contains every graph in $\mathcal{F}$ as a (not necessarily induced) subgraph. For the family of all graphs on $n$ vertices and of maximum degree at most…

We consider the statistics of extreme eigenvalues of random $d$-regular graphs, with $N^{\mathfrak c}\leq d\leq N^{1/3-{\mathfrak c}}$ for arbitrarily small ${\mathfrak c}>0$. We prove that in this regime, the fluctuations of extreme…

We prove that for every integer $d \ge 3$, the median eigenvalues of any graph of maximum degree $d$ are bounded above by $\sqrt{d-1}$. We also prove that, in three separate cases, the median eigenvalues of a graph of maximum degree $d$ are…

A signed graph $\Sigma = (G, \sigma)$ is a graph where the function $\sigma$ assigns either $1$ or $-1$ to each edge of the simple graph $G$. The adjacency matrix of $\Sigma$, denoted by $A(\Sigma)$, is defined canonically. In a recent…

We show that for sufficiently large $d$ and for $t\geq d+1$, there is a graph $G$ with average degree $(1-\varepsilon)\lambda t \sqrt{\ln d}$ such that almost every graph $H$ with $t$ vertices and average degree $d$ is not a minor of $G$,…

Given a family ${\cal F}$ of graphs, and a positive integer $n$, the Tur\'an number $ex(n,{\cal F})$ of ${\cal F}$ is the maximum number of edges in an $n$-vertex graph that does not contain any member of ${\cal F}$ as a subgraph. The order…

Let $d\geq 3$ be fixed and $G$ be a large random $d$-regular graph on $n$ vertices. We show that if $n$ is large enough then the entry distribution of every almost eigenvector $v$ of $G$ (with entry sum 0 and normalized to have length…

We give a new recurrence formula for the eigenvalues of the derangement graph. Consequently, we provide a simpler proof of the Alternating Sign Property of the derangement graph. Moreover, we prove that the absolute value of the eigenvalue…

For a graph G and integer r \geq 1 we denote the family of independent r-sets of V(G) by I^{(r)}(G). A graph G is said to be r-EKR if no intersecting subfamily of I^{(r)}(G) is larger than the largest such family all of whose members…

The skewness of a graph G is the minimum number of edges in G whose removal results in a planar graph. By appropriately introducing a weight to each edge of a graph, we determine, among other thing, the skewness of the generalized Petersen…

We consider random sub-graphs of a fixed graph $G=(V,E)$ with large minimum degree. We fix a positive integer $k$ and let $G_k$ be the random sub-graph where each $v\in V$ independently chooses $k$ random neighbors, making $kn$ edges in…

The zero locus of a function f on a graph G is defined as the graph with vertex set consisting of all complete subgraphs of G, on which f changes sign and where x,y are connected if one is contained in the other. For d-graphs, finite simple…

We give lower bounds on the maximum possible girth of an $r$-uniform, $d$-regular hypergraph with at most $n$ vertices, using the definition of a hypergraph cycle due to Berge. These differ from the trivial upper bound by an absolute…