Related papers: Eigenvalues of Maximal Abelian Covers
In the framework of the conjectured duality relation between large $N$ gauge theory and supergravity the spectra of masses in large $N$ gauge theory can be determined by solving certain eigenvalue problems in supergravity. In this paper we…
Given two hyperbolic curves over p-adic local fields, the absolute anabelian conjecture claims that any isomorphism between their \'etale fundamental group comes from an isomorphism of schemes. This conjecture was proven by S. Mochizuki for…
Let $A(G)$ and $D(G)$ be the adjacency and degree matrices of a simple graph $G$ on $n$ vertices, respectively. The \emph{$A_\alpha$-spectral radius} of $G$ is the largest eigenvalue of $A_\alpha (G)=\alpha D(G)+(1-\alpha)A(G)$ for a real…
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…
It is well known that spectral Tur\'{a}n type problem is one of the most classical {problems} in graph theory. In this paper, we consider the spectral Tur\'{a}n type problem. Let $G$ be a graph and let $\mathcal{G}$ be a set of graphs, we…
We consider the spectral structure of indefinite second order boundary-value problems on graphs. A variational formulation for such boundary-value problems on graphs is given and we obtain both full and half-range completeness results. This…
We show that the eigenpolytopes of graphs are universal in the sense that every polytope, up to affine equivalence, appears as the eigenpolytope of some positively weighted graph. We next extend the theory of graphical designs, which are…
It is known that the canonical double cover of any connected nonbipartite graph have an automorphism group of the form $H \rtimes \mathbb{Z}_2$, where $H$ is the set of automorphism which preserve bipartite parts. We construct connected…
We study the eigenvalues of the unique connected anti-regular graph $A_n$. Using Chebyshev polynomials of the second kind, we obtain a trigonometric equation whose roots are the eigenvalues and perform elementary analysis to obtain an…
In this paper we investigate Abel maps on normal surface singularities described in \cite{NNI}. We investigate the affine version of the class of the images of Abel maps on normal surface singularities. More precisely we consider the…
In recent work on equiangular lines, Jiang, Tidor, Yuan, Zhang, and Zhao showed that a connected bounded degree graph has sublinear second eigenvalue multiplicity. More generally they show that there cannot be too many eigenvalues near the…
An new eigenvalue $\mathbb R$-linear problem arisen in the theory of metamaterials is stated and constructively investigated for circular non-overlapping inclusions. An asymptotic formula for eigenvalues is deduced when the radii of…
The main purpose of this paper is to prove the following theorem on the defect relations for ample divisors of abelian varieties. Main Theorem. Let $A$ be an abelian variety of complex dimension $n$ and $D$ be an ample divisor in $A$. Let…
We analyze the Brill-Noether theory of trivalent graphs and multigraphs having largest possible automorphism group in a fixed genus. For trivalent multigraphs with loops of genus at least 3, we show that there exists a graph with maximal…
We derive upper bounds for the eigenvalues of the Kirchhoff Laplacian on a compact metric graph depending on the graph's genus g. These bounds can be further improved if $g = 0$, i.e. if the metric graph is planar. Our results are based on…
Let $G$ be a graph of order $n$, and let $A(G)$ and $D(G)$ be the adjacency matrix and the degree matrix of $G$ respectively. Define the convex linear combinations $A_\alpha (G)$ of $A (G)$ and $D (G) $ by $$A_\alpha (G)=\alpha…
We present a new approach to showing that random graphs are nearly optimal expanders. This approach is based on recent deep results in combinatorial group theory. It applies to both regular and irregular random graphs. Let G be a random…
We provide counterexamples to several conjectures concerning strongly maximal and strongly minimal structures in infinite graphs and hypergraphs. In particular, we construct 3-uniform hypergraphs without strongly maximal matchings and…
A graph $G$ is $[a,b]$-covered if for each edge $e$ of $G$ there is an $[a,b]$-factor containing it. For $a=b=1$, an $[a,b]$-covered graph is a matching covered graph. The structural theory of matching covered graphs constitutes a…
Let $\mathscr{G}_{n,\beta}$ be the set of graphs of order $n$ with given matching number $\beta$. Let $D(G)$ be the diagonal matrix of the degrees of the graph $G$ and $A(G)$ be the adjacency matrix of the graph $G$. The largest eigenvalue…