Related papers: Median eigenvalues of subcubic graphs

Recently, Ferrulli-Laptev-Safronov (2016arXiv161205304F) obtained eigenvalue estimates for an operator associated to bilayer graphene in terms of $L^q$ norms of the (possibly non-selfadjoint) potential. They proved that for $1<q<4/3$ all…

The Hadwiger number of a graph $G$, denoted by $h(G)$, is the order of the largest complete minor of $G$. A graph is said to be self-complementary if it is isomorphic to its complement. We prove that for all $n\equiv 0,1 (\text{mod 4})$ and…

In tensor eigenvalue problems, one is likely to be more interested in H-eigenvalues of tensors. The largest H-eigenvalue of a nonnegative tensor or of a uniform hypergraph is the spectral radius of the tensor or of the uniform hypergraph.…

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 inhomogeneous Erd\H{o}s-R\'enyi graphs. We suppose that the maximal mean degree $d$ satisfies $d \ll \log n$. We characterize the asymptotic behavior of the $n^{1 - o(1)}$ largest eigenvalues of the adjacency matrix and its…

The scissors congruence conjecture for the unimodular group is an analogue of Hilbert's third problem, for the equidecomposability of polytopes. Liu and Osserman studied the Ehrhart quasi-polynomials of polytopes naturally associated to…

For a graph $G$, we associate a family of real symmetric matrices, $S(G)$, where for any $A\in S(G)$, the location of the nonzero off-diagonal entries of $A$ are governed by the adjacency structure of $G$. Let $q(G)$ be the minimum number…

Let $(G,w)$ be a weighted graph with a weight-function $w: E(G)\to \mathbb R\backslash\{0\}$. A weighted graph $(G,w)$ is invertible to a new weighted graph if its adjacency matrix is invertible. A graph inverse has combinatorial interest…

The well-known 1-2-3 Conjecture asserts that the edges of every graph without an isolated edge can be weighted with $1$, $2$ and $3$ so that adjacent vertices receive distinct weighted degrees. This is open in general. We prove that every…

The maximal graph Dirichlet problem asks whether there exists a spacelike graph, in a semi-Euclidean space, with a given boundary and with mean curvature everywhere zero. We prove the existence of solutions to this problem under certain…

We investigate properties of signed graphs that have few distinct eigenvalues together with a symmetric spectrum. Our main contribution is to determine all signed $(0,2)$-graphs with vertex degree at most $6$ that have precisely two…

In this short note we study two questions about the existence of subgraphs of the hypercube $Q_n$ with certain properties. The first question, due to Erd\H{o}s--Hamburger--Pippert--Weakley, asks whether there exists a bounded degree…

Let G be a random subgraph of the n-cube where each edge appears randomly and independently with probability p. We prove that the largest eigenvalue of the adjacency matrix of G is almost surely \lambda_1(G)= (1+o(1))…

We present a new method for upper bounding the second eigenvalue of the Laplacian of graphs. Our approach uses multi-commodity flows to deform the geometry of the graph; we embed the resulting metric into Euclidean space to recover a bound…

Erd\H{o}s, Fajtlowicz and Staton asked for the least integer $f(k)$ such that every graph with more than $f(k)$ vertices has an induced regular subgraph with at least $k$ vertices. Here we consider the following relaxed notions. Let $g(k)$…

Markoff mod-$p$ graphs are conjectured to be connected for all primes $p$. In this paper, we use results of Chen and Bourgain, Gamburd, and Sarnak to confirm the conjecture for all $p > 3.448\cdot10^{392}$. We also provide a method that…

In this paper, we investigate the degree of $h$-polynomials of edge ideals of finite simple graphs. In particular, we provide combinatorial formulas for the degree of the $h$-polynomial for various fundamental classes of graphs such as…

Let $G$ be a graph and let $g, f$ be nonnegative integer-valued functions defined on $V(G)$ such that $g(v) \le f(v)$ and $g(v) \equiv f(v) \pmod{2}$ for all $v \in V(G)$. A $(g,f)$-parity factor of $G$ is a spanning subgraph $H$ such that…

For a real c \geq 1 and an integer n, let f(n,c) denote the maximum integer f so that every graph on n vertices contains an induced subgraph on at least f vertices in which the maximum degree is at most c times the minimum degree. Thus, in…

In this note we elaborate on some notions of surface area for discrete graphs which are closely related to the inverse degree. These notions then naturally lead to associated connectivity measures of graphs and to the definition of a…