Some new bounds on the spectral radius of graphs (original) (raw)

Sharp upper bounds on the spectral radius of graphs

Linear Algebra and its Applications, 2004

Let G be a simple connected graph with n vertices, m edges and degree sequence: d 1 d 2 · · · d n . The spectral radius ρ(G) of graph G is the largest eigenvalue of its adjacency matrix. In this paper, we present some sharp upper bounds of the spectral radius in terms of the degree sequence of graphs.

The new upper bounds on the spectral radius of weighted graphs

Applied Mathematics and Computation, 2012

We consider weighted graphs, where the edge weights are positive definite matrices. The eigenvalues of a graph are the eigenvalues of its adjacency matrix. We obtain another upper bound which is sharp on the spectral radius of the adjacency matrix and compare with some known upper bounds with the help of some examples of graphs. We also characterize graphs for which the bound is attained.

Bounds On The Second Stage Spectral Radius Of Graphs

2009

Let G be a graph of order n. The second stage adjacency matrix of G is the symmetric n × n matrix for which the ijth entry is 1 if the vertices vi and vj are of distance two; otherwise 0. The sum of the absolute values of this second stage adjacency matrix is called the second stage energy of G. In this paper we investigate a few properties and determine some upper bounds for the largest eigenvalue.

On the Aα-spectral radius of connected graphs

Ars Mathematica Contemporanea, 2022

For a simple graph G, the generalized adjacency matrix A α (G) is defined as A α (G) = αD(G) + (1 − α)A(G), α ∈ [0, 1], where A(G) is the adjacency matrix and D(G) is the diagonal matrix of the vertex degrees. It is clear that A 0 (G) = A(G) and 2A 1 2 (G) = Q(G) implying that the matrix A α (G) is a generalization of the adjacency matrix and the signless Laplacian matrix. In this paper, we obtain some new upper and lower bounds for the generalized adjacency spectral radius λ(A α (G)), in terms of vertex degrees, average vertex 2-degrees, the order, the size, etc. The extremal graphs attaining these bounds are characterized. We will show that our bounds are better than some of the already known bounds for some classes of graphs. We derive a general upper bound for λ(A α (G)), in terms of vertex degrees and positive real numbers b i. As application, we obtain some new upper bounds for λ(A α (G)). Further, we obtain some relations between clique number ω(G), independence number γ(G) and the generalized adjacency eigenvalues of a graph G.

Improved upper and lower bounds for the spectral radius of digraphs

Applied Mathematics and Computation, 2010

Let G be a digraph with n vertices and m arcs without loops and multiarcs. The spectral radius qðGÞ of G is the largest eigenvalue of its adjacency matrix. In this paper, sharp upper and lower bounds on qðGÞ are given. We show that some known bounds can be obtained from our bounds.

Sharp upper bounds on the distance spectral radius of a graph

Linear Algebra and its Applications, 2013

Let M = (m ij ) be a nonnegative irreducible n × n matrix with diagonal entries 0. The largest eigenvalue of M is called the spectral radius of the matrix M, denoted by ρ(M). In this paper, we give two sharp upper bounds of the spectral radius of matrix M. As corollaries, we give two sharp upper bounds of the distance matrix of a graph.

Bounds for the spectral radius of a graph when nodes are removed

Linear Algebra and its Applications, 2012

We present a new type of lower bound for the spectral radius of a graph in which m nodes are removed. As a corollary, Cioabȃ's theorem [4], which states that the maximum normalized principal eigenvector component in any graph never exceeds 1 √ 2 (with equality for the star), appears as a special case of our more general result.

A lower bound for the spectral radius of graphs with fixed diameter

European Journal of Combinatorics, 2010

We determine a lower bound for the spectral radius of a graph in terms of the number of vertices and the diameter of the graph. For the specific case of graphs with diameter three we give a slightly better bound. We also construct families of graphs with small spectral radius, thus obtaining asymptotic results showing that the bound is of the right order. We also relate these results to the extremal degree/diameter problem.

On the generalized adjacency spectral radius of digraphs

Linear and Multilinear Algebra

Let D be a digraph of order n and let A(D) be the adjacency matrix of D. Let Deg(D) be the diagonal matrix of vertex out-degrees of D. For any real α ∈ [0, 1], the generalized adjacency matrix A α (D) of the digraph D is defined as A α (D) = αDeg(D) + (1 − α)A(D). The largest modulus of the eigenvalues of A α (D) is called the generalized adjacency spectral radius or the A α-spectral radius of D. In this paper, we obtain some lower bounds for the spectral radius of A α (D) in terms of the number of vertices, the number of arcs and the number of closed walks of the digraph D. The extremal graphs attained these lower bounds are determined. We also obtain some bounds for the spectral radius of A α (D) in terms of the vertex out-degrees, the vertex average 2-out-degrees of the vertices of D and parameter α. We characterize the extremal digraphs attaining these bounds.