Related papers: Explicit bounds from the Alon-Boppana theorem

From Alon and Boppana, and Serre, we know that for any given integer $k\geq 3$ and real number $\lambda<2\sqrt{k-1}$, there are finitely many $k$-regular graphs whose second largest eigenvalue is at most $\lambda$. In this paper, we…

Let $v(k, \lambda)$ be the maximum number of vertices of a connected $k$-regular graph with second largest eigenvalue at most $\lambda$. The Alon-Boppana Theorem implies that $v(k, \lambda)$ is finite when $k > \frac{\lambda^2 + 4}{4}$. In…

The Alon-Boppana theorem confirms that for every $\epsilon>0$ and every integer $d\ge3$, there are only finitely many $d$-regular graphs whose second largest eigenvalue is at most $2\sqrt{d-1}-\epsilon$. Serre gave a strengthening showing…

For any $\epsilon > 0$, we show that if $G$ is a regular graph on $n \gg_\epsilon 1$ vertices that is $\epsilon$-far (differs by at least $\epsilon n^2$ edges) from any Tur\'{a}n graph, then its second eigenvalue $\lambda_2$ satisfies…

We prove a general upper bound on the $k$-th adjacency eigenvalue of a graph. For $k\ge 2$, we show that \[ \lambda_k(G)\le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1 \] for every graph $G$ on $n$ vertices. We build on a recent approach that…

Lower bounds for the first and the second eigenvalue of uniform hypergraphs which are regular and linear are obtained. One of these bounds is a generalization of the Alon-Boppana Theorem to hypergraphs.

Let $b(k,\ell,\theta)$ be the maximum number of vertices of valency $k$ in a $(k,\ell)$-semiregular bipartite graph with second largest eigenvalue $\theta$. We obtain an upper bound for $b(k,\ell,\theta)$ for $0 < \theta < \sqrt{k-1} +…

For a given complex square matrix $A$ with constant row sum, we establish two new eigenvalue inclusion sets. Using these bounds, first we derive bounds for the second largest and smallest eigenvalues of adjacency matrices of $k$-regular…

The unraveled ball of radius $r$ centered at a vertex $v$ in a weighted graph $G$ is the ball of radius $r$ centered at $v$ in the universal cover of $G$. We present a general bound on the maximum spectral radius of unraveled balls of fixed…

A celebrated result of Alon from 1993 states that any $d$-regular graph on $n$ vertices (where $d=O(n^{1/9})$) has a bisection with at most $\frac{dn}{2}(\frac{1}{2}-\Omega(\frac{1}{\sqrt{d}}))$ edges, and this is optimal. Recently, this…

The maximum number of vertices in a graph of maximum degree $\Delta\ge 3$ and fixed diameter $k\ge 2$ is upper bounded by $(1+o(1))(\Delta-1)^{k}$. If we restrict our graphs to certain classes, better upper bounds are known. For instance,…

We prove that the number of vertices of a polytope of a particular kind is exponentially large in the dimension of the polytope. As a corollary, we prove that an n-dimensional centrally symmetric polytope with O(n) facets has 2^{Omega(n)}…

Let $G$ be a graph on $n \ge 3$ vertices, whose adjacency matrix has eigenvalues $\lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_n$. The problem of bounding $\lambda_k$ in terms of $n$ was first proposed by Hong and was studied by Nikiforov,…

We study the maximum number of straight-line segments connecting $n$ points in convex position in the plane, so that each segment intersects at most $k$ others. This question can also be framed as the maximum number of edges of an outer…

Let $b(k,\theta)$ be the maximum order of a connected bipartite $k$-regular graph whose second largest eigenvalue is at most $\theta$. In this paper, we obtain a general upper bound for $b(k,\theta)$ for any $0\leq \theta< 2\sqrt{k-1}$. Our…

Assume that $X$ is a connected $(q+1)$-regular undirected graph of finite order $n$. Let $A$ denote the adjacency matrix of $X$. Let $\lambda_1=q+1>\lambda_2\geq \lambda_3\geq \ldots \geq \lambda_n$ denote the eigenvalues of $A$. The…

In this paper, we study the order of a maximal clique in an amply regular graph with a fixed smallest eigenvalue by considering a vertex that is adjacent to some (but not all) vertices of the maximal clique. As a consequence, we show that…

Let $k \geq 3$. We prove the following three bounds for the matching number, $\alpha'(G)$, of a graph, $G$, of order $n$ size $m$ and maximum degree at most $k$. If $k$ is odd, then $\alpha'(G) \ge \left( \frac{k-1}{k(k^2 - 3)} \right) n \,…

In this article we generalize a theorem of Benson for generalized quadrangles to strongly regular graphs and directed strongly regular graphs. The main result provides numerical restrictions on the number of fixed vertices and the number of…

A topological graph is $k$-quasi-planar if it does not contain $k$ pairwise crossing edges. A 20-year-old conjecture asserts that for every fixed $k$, the maximum number of edges in a $k$-quasi-planar graph on $n$ vertices is $O(n)$. Fox…