Related papers: A spectral Erdos-Stone-Bollobas theorem
Let $G$ be a graph. We say that a hypergraph $H$ is a Berge-$G$ if there is a bijection $\phi: E(G)\to E(H)$ such that $e\subseteq \phi(e)$ for all $e\in E(G)$. For any $r$-uniform hypergraph $H$ and a real number $p\geq 1$, the…
Write $\rho\left( G\right) $ for the spectral radius of a graph $G$ and $S_{n,r}$ for the join $K_{r}\vee\overline{K}_{n-r}.$ Let $n>r\geq2$ and $G$ be a $K_{r+1}$-saturated graph of order $n.$ Recently Kim, Kim, Kostochka, and O determined…
The celebrated Brown-Erd\H{o}s-S\'os conjecture states that for every fixed $e$, every $3$-uniform hypergraph with $\Omega(n^2)$ edges contains $e$ edges spanned by $e+3$ vertices. Up to this date all the approaches towards resolving this…
It is known that the spectral radius of a digraph with k edges is \le \sqrt{k}, and that this inequality is strict except when k is a perfect square. For k=m^2 + \ell, \ell fixed, m large, Friedland showed that the optimal digraph is…
A recent framework for generalizing the Erdos-Ko-Rado Theorem, due to Holroyd, Spencer, and Talbot, defines the Erdos-Ko-Rado property for a graph in terms of the graph's independent sets. Since the family of all independent sets of a graph…
We prove that for any weakly convergent sequence of finite graphs with bounded vertex degrees, there exists a topological limit graphing.
In this paper, we study the higher Steklov eigenvalues of graphs on surfaces. We obtain the upper bound of higher Steklov eigenvalues of a finite graph $G$ with boundary $B$ and genus $g$ by using metrical deformation via probability flows.…
We show that the spectral-gap of a general zero range process can be controlled in terms of the spectral-gap of a single particle. This is in the spirit of Aldous' famous spectral-gap conjecture for the interchange process. Our main…
We characterise the form of all simple, finite graphs for which the girth of the graph is equal to the circumference of the graph. We apply this to prove a bound on the number of edges in such a graph.
A coupling method and an analytic one allow us to prove new lower bounds for the spectral gap of reversible diffusions on compact manifolds. Those bounds are based on the a notion of curvature of the diffusion, like the coarse Ricci…
We give upper and lower bounds for the spectral radius of a nonnegative matrix by using its average 2-row sums, and characterize the equality cases if the matrix is irreducible. We also apply these bounds to various nonnegative matrices…
The core of this note is the observation that links between circle packings of graphs and potential theory developed in \cite{BeSc01} and \cite{HS} can be extended to higher dimensions. In particular, it is shown that every limit of finite…
We prove a central limit theorem for a certain class of functions on sparse rank-one inhomogeneous random graphs endowed with additional i.i.d. edge and vertex weights. Our proof of the central limit theorem uses a perturbative form of…
We propose an approach to calculate the critical percolation threshold for finite-sized Erdos-Renyi digraphs using minimal Hamiltonian cycles. We obtain an analytically exact result, valid non-asymptotically for all graph sizes, which…
We give a bound for the graph energy with given maximal degree in terms of the second and fourth moments of a graph. In the case in which the graph is $d$-regular we obtain the bound that is given in Van Dam, E. et al. (2014). through…
We present a quantum ergodicity theorem for fixed spectral window and sequences of compact hyperbolic surfaces converging to the hyperbolic plane in the sense of Benjamini and Schramm. This addresses a question posed by Colin de…
Let $t\geq3$ and $G$ be a graph of order $n,$ with no $K_{2,t}$ minor. If $n>400t^{6}$, then the spectral radius $\mu\left( G\right) $ satisfies \[ \mu\left( G\right) \leq\frac{t-1}{2}+\sqrt{n+\frac{t^{2}-2t-3}{4}}, \] with equality if and…
Shiu, Chan and Chang [On the spectral radius of graphs with connectivity at most $k$, J. Math. Chem., 46 (2009), 340-346] studied the spectral radius of graphs of order $n$ with $\kappa(G) \leq k$ and showed that among those graphs, the…
Some arithmetic properties of spectral curves are discussed: the spectral curve, for example, of a charge $n\ge2$ Euclidean BPS monopole is not defined over $\overline{\mathbb{Q}}$ if smooth.
We prove that for every graph $G$ with $n$ vertices, the treewidth of $G$ plus the treewidth of the complement of $G$ is at least $n-2$. This bound is tight.