Related papers: Quasirandom Cayley graphs
We study geometric and topological properties of infinite graphs that are quasi-isometric to a planar graph of bounded degree. We prove that every locally finite quasi-transitive graph excluding a minor is quasi-isometric to a planar graph…
One of the main questions that arise when studying random and quasi-random structures is which properties P are such that any object that satisfies P "behaves" like a truly random one. In the context of graphs, Chung, Graham, and Wilson…
We generalise the standard constructions of a Cayley graph in terms of a group presentation by allowing some vertices to obey different relators than others. The resulting notion of presentation allows us to represent every vertex…
In this note, we show that locally finite quasi-transitive graphs are quasi-isometric to trees if and only if every other locally finite quasi-transitive graph quasi-isometric to them is minor excluded. This generalizes results by…
It is proven following [18[ that Laplacians with standard vertex continuous on metric trees and with standard and Dirichlet conditions on arbitrary metric graphs possess an infinite sequence of simple eigenvalues with the eigenfunctions not…
We consider nonregular graphs having precisely three distinct eigenvalues. The focus is mainly on the case of graphs having two distinct valencies and our results include constructions of new examples, structure theorems, valency…
For a given graph $F$ we consider the family of (finite) graphs $G$ with the Ramsey property for $F$, that is the set of such graphs $G$ with the property that every two-colouring of the edges of $G$ yields a monochromatic copy of $F$. For…
We study continuous-time quantum walks on normal Cayley graphs of certain non-abelian groups, called extraspecial groups. By applying general results for graphs in association schemes we determine the precise conditions for perfect state…
We study spectral properties of the standard (also called Kirchhoff) Laplacian and the anti-standard (or anti-Kirchhoff) Laplacian on a finite, compact metric graph. We show that the positive eigenvalues of these two operators coincide…
We prove that every locally finite, quasi-transitive graph with a thick end whose cycle space is generated by cycles of bounded length contains the full-grid as an asymptotic minor and as a diverging minor. This in particular includes all…
For every fixed graph $H$ and every fixed $0 < \alpha < 1$, we show that if a graph $G$ has the property that all subsets of size $\alpha n$ contain the ``correct'' number of copies of $H$ one would expect to find in the random graph…
Chung and Graham began the systematic study of k-uniform hypergraph quasirandom properties soon after the foundational results of Thomason and Chung-Graham-Wilson on quasirandom graphs. One feature that became apparent in the early work on…
In this paper, we study self-complementary distance-regular Cayley graphs over abelian groups. We prove that if a regular graph is self-complementary distance-regular, then it is self-complementary strongly regular. We also deal with…
This paper deals with some of the algebraic properties of Sierpi\'nski graphs and a family of regular generalized Sierpi\'nski graphs. For the family of regular generalized Sierpi\'nski graphs, we obtain their spectrum and characterize…
In this paper we prove the semi-circular law for the eigenvalues of regular random graph $G_{n,d}$ in the case $d\rightarrow \infty$, complementing a previous result of McKay for fixed $d$. We also obtain a upper bound on the infinity norm…
The transmission problem is a system of two second-order elliptic equations of two unknowns equipped with the Cauchy data on the boundary. After four decades of research motivated by scattering theory, the spectral properties of this…
The Kohayakawa-Nagle-R\"odl-Schacht conjecture roughly states that every sufficiently large locally $d$-dense graph $G$ on $n$ vertices must contain at least $(1-o(1))d^{|E(H)|}n^{|V(H)|}$ copies of a fixed graph $H$. Despite its important…
This paper represents a significant leap forward in the problem of enumerating vertex-transitive graphs. Recent breakthroughs on symmetry of Cayley (di)graphs show that almost all finite Cayley (di)graphs have the smallest possible…
In this paper we show that the equivalences between certain properties of closed subanalytic sets proved by E. Bierstone and P. Milman in \cite{[BM-1]} hold for closed sets definable in quasianalytic o-minimal structures. In particular we…
Let $\Gamma$ be a finite group acting transitively on $[n]=\{1,2,\ldots,n\}$, and let $G=\mathrm{Cay}(\Gamma,T)$ be a Cayley graph of $\Gamma$. The graph $G$ is called normal if $T$ is closed under conjugation. In this paper, we obtain an…