Related papers: Eigenvalue problems of Nordhaus-Gaddum type

In 1956, Nordhaus and Gaddum gave lower and upper bounds on the sum and the product of the chromatic number of a graph and its complement, in terms of the order of the graph. Since then, any bound on the sum and/or the product of an…

We give inequalities relating the eigenvalues of the adjacency matrix and the Laplacian of a graph, and its minimum and maximum degrees. The results are applied to derive new conditions for quasi-randomness of graphs.

For a graph $G$ on $n$ vertices with normalized Laplacian eigenvalues $0 = \lambda_1(G) \leq \lambda_2(G) \leq \cdots \leq \lambda_n(G)$ and graph complement $G^c$, we prove that \begin{equation*} \max\{\lambda_2(G),\lambda_2(G^c)\}\geq…

Let $G$ be a graph on $n$ vertices, with complement $\overline{G}$. The spectral gap of the transition probability matrix of a random walk on $G$ is used to estimate how fast the random walk becomes stationary. We prove that the larger…

Let $G$ be an $n$-vertex simple graph with adjacency matrix $A_G$. The \emph{complement rank} of $G$ is defined as $\operatorname{rank}(A_G+I)$, where $I$ is the identity matrix. In this paper we study Nordhaus--Gaddum type bounds for the…

For a fixed positive integer $t$, we consider the graph colouring problem in which edges at distance at most $t$ are given distinct colours. We obtain sharp lower bounds for the distance-$t$ chromatic index, the least number of colours…

Let $G$ be a graph, and let $\lambda(G)$ denote the smallest eigenvalue of $G$. First, we provide an upper bound for $\lambda(G)$ based on induced bipartite subgraphs of $G$. Consequently, we extract two other upper bounds, one relying on…

Let F(G) be a fixed linear combination of the k extremal eigenvalues of a graph G and of its complement. The problem of finding max{F(G):v(G)=n} generalizes a number of problems raised previously in the literature. We show that the limit…

Let $G$ be a connected uniform hypergraphs with maximum degree $\Delta$, spectral radius $\lambda$ and minimum H-eigenvalue $\mu$. In this paper, we give some lower bounds for $\Delta-\lambda$, which extend the result of [S.M. Cioab\u{a},…

We derive bounds on the size of an independent set based on eigenvalues. This generalizes a result due to Delsarte and Hoffman. We use this to obtain new bounds on the independence number of the Erd\H{o}s-R\'{e}nyi graphs. We investigate…

Let $G$ be a connected undirected graph with $n$, $n\ge 3$, vertices and $m$ edges. Denote by $\rho_1 \ge \rho_2 \ge \cdots > \rho_n =0$ the normalized Laplacian eigenvalues of $G$. Upper and lower bounds of $\rho_i$, $i=1,2,\ldots , n-1$,…

The spectral radius of a graph is the spectral radius of its adjacency matrix. A threshold graph is a simple graph whose vertices can be ordered as $v_1, v_2, \ldots, v_n$, so that for each $2 \le i \le n$, vertex $v_i$ is either adjacent…

Let A(G) be the adjacency tensor (hypermatrix) of uniform hypergraph G. The maximum modulus of the eigenvalues of A(G) is called the spectral radius of G. In this paper, the conjecture of Fan et al. in [5] related to compare the spectral…

A dominating set in a graph $G$ is a set $S$ of vertices such that every vertex of $G$ is either in $S$ or is adjacent to a vertex in $S$. Nordhaus-Gaddum inequailties relate a graph $G$ to its complement $\bar{G}$. In this spirit Wagner…

The spectral radius {\rho}(G) of a digraph G is the maximum modulus of the eigenvalues of its adjacency matrix. We present bounds on {\rho}(G) that are often tighter and are applicable to a larger class of digraphs than previously reported…

Let $G$ be an irregular graph on $n$ vertices with maximum degree $\Delta$ and diameter $D$. We show that \Delta-\lambda_1>\frac{1}{nD} where $\lambda_1$ is the largest eigenvalue of the adjacency matrix of $G$. We also study the effect of…

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…

In this paper, we obtain the sharp upper and lower bounds for the spectral radius of a nonnegative irreducible matrix. We also apply these bounds to various matrices associated with a graph or a digraph, obtain some new results or known…

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…

The extended adjacency matrix of a graph with $n$ vertices is a real symmetric matrix of order $n\times n$ whose $(i,j)$-th entry is the average of the ratio of the degree of the vertex $i$ to that of the vertex $j$ and its reciprocal when…