Related papers: Periodic quantum graphs with predefined spectral g…
For a non-negative integer $s\le |V(G)|-3$, a graph $G$ is $s$-Hamiltonian if the removal of any $k\le s$ vertices results in a Hamiltonian graph. Given a connected simple graph $G$ that is not isomorphic to a path, a cycle, or a $K_{1,3}$,…
Following recent work by Koll\'{a}r and Sarnak, we study gaps in the spectra of large connected cubic and quartic graphs with minimum spectral gap. We focus on two sequences of graphs, denoted $\Delta_n$ and $\Gamma_n$ which are more…
We consider admissible random walks on hyperbolic graphs. For a given harmonic function on such a graph, we prove that asymptotic properties of non-tangential boundedness and non-tangential convergence are almost everywhere equivalent. The…
In this study we consider the $\Gamma$-limit of a highly oscillatory Riemannian metric length functional as its period tends to 0. The metric coefficient takes values in either $\{1,\infty\}$ or $\{1,\beta \varepsilon^{-p}\}$ where…
We show that under certain conditions the square of the graph obtained by identifying a vertex in two graphs with hamiltonian square is also hamiltonian. Using this result, we prove necessary and sufficient conditions for hamiltonicity of…
We consider a family of open sets $M_\epsilon$ which shrinks with respect to an appropriate parameter $\epsilon$ to a graph. Under the additional assumption that the vertex neighbourhoods are small we show that the appropriately shifted…
Quantum graphs are defined by having a Laplacian defined on the edges of a metric graph with boundary conditions on each vertex such that the resulting operator, $\mathbf{L}$, is self-adjoint. We use Neumann boundary conditions although we…
We uncover the very rich graph topology of generic bounded non-Hermitian spectra, distinct from the topology of conventional band invariants and complex spectral winding. The graph configuration of complex spectra are characterized by the…
This paper proposes to broaden the canonical formulation of quantum mechanics. Ordinarily, one imposes the condition $H^\dagger=H$ on the Hamiltonian, where $\dagger$ represents the mathematical operation of complex conjugation and matrix…
Let $\Gamma$ be a finite, undirected, connected, simple graph. We say that a matching $\mathcal{M}$ is a \textit{permutable $m$-matching} if $\mathcal{M}$ contains $m$ edges and the subgroup of $\text{Aut}(\Gamma)$ that fixes the matching…
Consider a random graph process where vertices are chosen from the interval $[0,1]$, and edges are chosen independently at random, but so that, for a given vertex $x$, the probability that there is an edge to a vertex $y$ decreases as the…
Dirac's classical theorem asserts that, for $n \ge 3$, any $n$-vertex graph with minimum degree at least $n/2$ is Hamiltonian. Furthermore, if we additionally assume that such graphs are regular, then, by the breakthrough work of Csaba,…
Quantum graphs have attracted attention from mathematicians for some time. A quantum graph is defined by having a Laplacian on each edge of a metric graph and imposing boundary conditions at the vertices to get an eigenvalue problem. A…
We present full description of spectra for a Hamiltonian defined on periodic hexagonal elastic lattices. These continua are constructed out of Euler-Bernoulli beams, each governed by a scalar-valued self-adjoint operator, which is also…
We consider Laplace operators on periodic discrete graphs perturbed by guides, i.e., graphs which are periodic in some directions and finite in other ones. The spectrum of the Laplacian on the unperturbed graph is a union of a finite number…
A graph $\Gamma$ is $G$-symmetric if it admits $G$ as a group of automorphisms acting transitively on the set of arcs of $\Gamma$, where an arc is an ordered pair of adjacent vertices. Let $\Gamma$ be a $G$-symmetric graph such that its…
A graph $G$ is said to be \emph{determined by its spectrum} if any graph having the same spectrum as $G$ is isomorphic to $G$. Let $K_n \setminus P_{\ell}$ be the graph obtained from $K_n$ by removing edges of $P_\ell$, where $P_\ell$ is a…
We show that every $(n,d,\lambda)$-graph contains a Hamilton cycle for sufficiently large $n$, assuming that $d\geq \log^{6}n$ and $\lambda\leq cd$, where $c=\frac{1}{70000}$. This significantly improves a recent result of Glock, Correia…
An \textit{\(m \times n\) grid graph} is the induced subgraph of the square lattice whose vertex set consists of all integer grid points \(\{(i,j) : 0 \leq i < m,\ 0 \leq j < n\}\). Let $H$ and $K$ be Hamiltonian cycles in an $m \times n$…
Let $\Gamma$ be a simple finite graph with vertex set $V(\Gamma)$ and edge set $E(\Gamma)$. Let $\mathcal{R}$ be an equivalence relation on $V(\Gamma)$. The $\mathcal{R}$-super $\Gamma$ graph $\Gamma^{\mathcal{R}}$ is a simple graph with…