Related papers: On high-girth expander graphs with localized eigen…

We prove improved bounds on how localized an eigenvector of a high girth regular graph can be, and present examples showing that these bounds are close to sharp. This study was initiated by Brooks and Lindenstrauss (2009) who relied on the…

We show that for every prime $d$ and $\alpha\in (0,1/6)$, there is an infinite sequence of $(d+1)$-regular graphs $G=(V,E)$ with girth at least $2\alpha \log_{d}(|V|)(1-o_d(1))$, second adjacency matrix eigenvalue bounded by…

The main purpose of the paper is to construct a sequence of graphs of constant degree with indefinitely growing girths admitting embeddings into $\ell_1$ with uniformly bounded distortions. This result answers the problem posed by N.…

We introduce a general class of algorithms and supply a number of general results useful for analysing these algorithms when applied to regular graphs of large girth. As a result, we can transfer a number of results proved for random…

We describe a new random greedy algorithm for generating regular graphs of high girth: Let $k\geq 3$ and $c \in (0,1)$ be fixed. Let $n \in \mathbb{N}$ be even and set $g = c \log_{k-1} (n)$. Begin with a Hamilton cycle $G$ on $n$ vertices.…

Using our previously published algorithm, we analyze the eigenvectors of the generalized Laplacian for two metric graphs occurring in practical applications. As expected, localization of an eigenvector is rare and the network should be…

Hypergraph expanders are hypergraphs with surprising, non-intuitive expansion properties. In a recent paper, the first author gave a simple construction, which can be randomized, of $3$-uniform hypergraph expanders with polylogarithmic…

Let $G=(V,E)$ be a finite graph. For $v\in V$ we denote by $G_v$ the subgraph of $G$ that is induced by $v$'s neighbor set. We say that $G$ is $(a,b)$-regular for $a>b>0$ integers, if $G$ is $a$-regular and $G_v$ is $b$-regular for every…

Although the spectral properties of random graphs have been a long-standing focus of network theory, the properties of right eigenvectors of directed graphs have so far eluded an exact analytic treatment. We present a general theory for the…

Expander graphs have been, during the last five decades, the subject of a most fruitful interaction between pure mathematics and computer science, with influence and applications going both ways (cf. [Lub94], [HLW06], [Lub12] and the…

We prove delocalization of eigenvectors of vertex-transitive graphs via elementary estimates of the spectral projector. We recover in this way known results which were formerly proved using representation theory. Similar techniques show…

Amply regular graphs are graphs with local distance-regularity constraints. In this paper, we prove a weaker version of a conjecture proposed by Qiao, Park, and Koolen on diameter bounds of amply regular graphs and make new progress on…

A graph $G$ covers a graph $H$ if there exists a locally bijective homomorphism from $G$ to $H$. We deal with regular covers in which this locally bijective homomorphism is prescribed by an action of a subgroup of ${\rm Aut}(G)$. Regular…

We improve recent results relating graph eigenvalues to other graph parameters like girth, domination number, and minimum degree.

The goal of this paper is to present a centrality measurement for the nodes of a hypergraph, by using existing literature which extends eigenvector centrality from a graph to a hypergraph, and literature which give a general centrality…

We construct and study a class of algebras associated to generalized layered graphs, i.e. directed graphs with a ranking function on their vertices. Each finite directed acyclic graph admits countably many structures of a generalized…

Extremal graph theory studies the maximum or minimum number of subgraphs isomorphic to a prescribed graph under given constraints. \textit{Localization} has recently emerged as a framework that refines such problems by assigning extremal…

We present new degree-sequence lower bounds on the expected size of an independent set from the hard-core model. For arbitrary graphs, we establish a multivariate lower bound inspired by a conjecture of the first author and Kang and a…

An amply regular graph is a regular graph such that any two adjacent vertices have $\alpha$ common neighbors and any two vertices with distance $2$ have $\beta$ common neighbors. We prove a sharp lower bound estimate for the Lin--Lu--Yau…

We show that every regular graph with good local expansion has a spanning Lipschitz subgraph with large girth and minimum degree. In particular, this gives a finite analogue of the dynamical solution to the von Neumann problem by Gaboriau…