Related papers: An inequality for regular near polygons

For $k\ge 1$, the $k$-independence number $\alpha_k$ of a graph is the maximum number of vertices that are mutually at distance greater than $k$. The well-known inertia and ratio bounds for the (1-)independence number $\alpha(=\alpha_1)$ of…

A $k$-proper edge-coloring of a graph G is called adjacent vertex-distinguishing if any two adjacent vertices are distinguished by the set of colors appearing in the edges incident to each vertex. The smallest value $k$ for which $G$ admits…

Fix an integer $n \geq 1$, and consider the set of all connected finite simple graphs on $n$ vertices. For each $G$ in this set, let $I(G)$ denote the edge ideal of $G$ in the polynomial ring $R = K[x_1,\ldots,x_n]$. We initiate a study of…

The parametric geometry of numbers has allowed to visualize the simultaneous approximation properties of a collection of real numbers through the combined graph of the related successive minima functions. Several inequalities among…

Let $G$ be a simple connected graph of order $n$ and $D(G)$ be the distance matrix of $G.$ Suppose that $\lambda_{1}(D(G))\geq\lambda_{2}(D(G))\geq\cdots\geq\lambda_{n}(D(G))$ are the distance spectrum of $G$. A graph $G$ is said to be…

In this paper, we show that the largest signless Laplacian H-eigenvalue of a connected $k$-uniform hypergraph $G$, where $k \ge 3$, reaches its upper bound $2\Delta(G)$, where $\Delta(G)$ is the largest degree of $G$, if and only if $G$ is…

In this paper we will look at the relationship between the intersection number c2 and its diameter for a distance-regular graph. And also, we give some tools to show that a distance-regular graph with large c2 is bipartite, and a tool to…

Let $\Gamma$ denote a bipartite distance-regular graph with diameter $D \ge 4$ and valency $k \ge 3$. Let $X$ denote the vertex set of $\Gamma$, and let $A$ denote the adjacency matrix of $\Gamma$. For $x \in X$ let $T=T(x)$ denote the…

Eigenvalues of a graph are the eigenvalues of the corresponding (0,1)-adjacency matrix. The second largest eigenvalue lambda_2 provides significant information on characteristics and structure of graphs. Therefore, finding bounds for…

Given an undirected graph $G$, the classical Cheeger constant, $h_G$, measures the optimal partition of the vertices into 2 parts with relatively few edges between them based upon the sizes of the parts. The well-known Cheeger's inequality…

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…

We first describe a system of inequalities (Horn's inequalities) that characterize eigenvalues of sums of Hermitian matrices. When we apply this system for integral Hermitian matrices, one can directly test it by using Littlewood-Richardson…

In this paper, all graphs whose adjacency matrix has at most two eigenvalues (multiplicities included) different from $2$ and $-1$ are determined. These graphs conclude a class of generalized friendship graphs $F_{t,r,k}, $ which is the…

For a connected graph $G$ with order $n$ and an integer $k\geq 1$, we denote by $$S_k(D(G))=\lambda_1(D(G))+\cdots+\lambda_k(D(G))$$ the sum of $k$ largest distance eigenvalues of $G$. In this paper, we consider the sharp upper bound and…

Bollob\'as and Nikiforov [J. Combin. Theory, Ser. B. 97 (2007) 859--865] conjectured the following. If $G$ is a $K_{r+1}$-free graph on at least $r+1$ vertices and $m$ edges, then $\lambda^2_1(G)+\lambda^2_2(G)\leq \frac{r-1}{r}\cdot2m$,…

This paper presents sufficient conditions for Hamiltonian paths and cycles in graphs. Letting $\lambda\left( G\right) $ denote the spectral radius of the adjacency matrix of a graph $G,$ the main results of the paper are: (1) Let $k\geq1,$…

Let $G$ be a random $d$-regular graph. We prove that for every constant $\alpha > 0$, with high probability every eigenvector of the adjacency matrix of $G$ with eigenvalue less than $-2\sqrt{d-2}-\alpha$ has $\Omega(n/$polylog$(n))$ nodal…

One of the best-known results in spectral graph theory is the inequality of Hoffman \[ \chi\left( G\right) \geq1-\frac{\lambda\left( G\right) }{\lambda_{\min }\left( G\right) }, \] where $\chi\left( G\right) $ is the chromatic number of a…

For a graph $G$ of order $n$, let $$ \lambda_1(G)\ge \cdots \ge \lambda_n(G) $$ be the eigenvalues of its adjacency matrix. We prove that every graph $G$ on $n\ge 3$ vertices satisfies $$ \lambda_3(G)\le \frac{n}{3}-1, $$ thereby solving a…

The parameter $q(G)$ of a graph $G$ is the minimum number of distinct eigenvalues over the family of symmetric matrices described by $G$. It is shown that the minimum number of edges necessary for a connected graph $G$ to have $q(G)=2$ is…