Conjectured bounds for the sum of squares of positive eigenvalues of a graph (original) (raw)

Nordhaus–Gaddum and other bounds for the sum of squares of the positive eigenvalues of a graph

Linear Algebra and its Applications, 2017

Terpai [22] proved the Nordhaus-Gaddum bound that µ(G) + µ(G) ≤ 4n/3 − 1, where µ(G) is the spectral radius of a graph G with n vertices. Let s + denote the sum of the squares of the positive eigenvalues of G. We prove that s + (G)+ s + (G) < √ 2n and conjecture that s + (G) + s + (G) ≤ 4n/3 − 1. We have used AutoGraphiX and Wolfram Mathematica to search for a counterexample. We also consider Nordhaus-Gaddum bounds for s + and bounds for the Randić index.

On a conjecture involving the second largest signless Laplacian eigenvalue and the index of graphs

2020

Let G be a simple connected graph of order n ≥ 9. Let q2 be the second largest signless Laplacian eigenvalue of G and λ1 be the index of G. Cvetković et al. [Publ. Inst. Math. (Beograd) 81(95) (2007) 11–27] conjectured that 1− √ n− 1 ≤ q2 − λ1 ≤ n− 2− √ 2n− 4, where the left equality holds if and only if G is the star K1,n−1, and the right equality holds if and only if G is the complete bipartite graph K2, n−2. Das [Linear Algebra Appl. 435 (2011) 2420–2424] proved that 1− √ n− 1 ≤ q2−λ1 and characterized the graphs attaining the equality. In this note, we prove that the inequality q2 − λ1 ≤ n − 2 − √ 2n− 4 holds for a certain class of graphs.

New Spectral Bounds on the Chromatic Number Encompassing all Eigenvalues of the Adjacency Matrix

The electronic journal of combinatorics

The purpose of this article is to improve existing lower bounds on the chromatic number χ. Let μ[subscript 1],…,μ[subscript n] be the eigenvalues of the adjacency matrix sorted in non-increasing order. First, we prove the lower bound χ ≥ 1 + max[subscript m]{∑[m over i=1]μ[subscript i]/ − ∑[m over i=1]μ[subscript n−i+1]} for m = 1,…,n − 1. This generalizes the Hoffman lower bound which only involves the maximum and minimum eigenvalues, i.e., the case m = 1. We provide several examples for which the new bound exceeds the Hoffman lower bound. Second, we conjecture the lower bound χ ≥ 1 + s[superscript +/s[superscript −], where s[superscript +] and s[superscript −] are the sums of the squares of positive and negative eigenvalues, respectively. To corroborate this conjecture, we prove the bound χ ≥ s[superscript +]/s[superscript −]. We show that the conjectured lower bound is true for several families of graphs. We also performed various searches for a counter-example, but none was foun...

Nonpositive eigenvalues of the adjacency matrix and lower bounds for Laplacian eigenvalues

Discrete Mathematics, 2013

Let NP O(k) be the smallest number n such that the adjacency matrix of any undirected graph with n vertices or more has at least k nonpositive eigenvalues. We show that NP O(k) is well-defined and prove that the values of NP O(k) for k = 1, 2, 3, 4, 5 are 1, 3, 6, 10, 16 respectively. In addition, we prove that for all k ≥ 5, R(k, k + 1) ≥ NP O(k) > T k , in which R(k, k + 1) is the Ramsey number for k and k + 1, and T k is the k th triangular number. This implies new lower bounds for eigenvalues of Laplacian matrices: the k-th largest eigenvalue is bounded from below by the NP O(k)-th largest degree, which generalizes some prior results.

Bounds for eigenvalues of a graph

Journal of Mathematical Inequalities, 2010

New lower bounds for eigenvalues of a simple graph are derived. Upper and lower bounds for eigenvalues of bipartite graphs are presented in terms of traces and degree of vertices. Finally a non-trivial lower bound for the algebraic connectivity of a connected graph is given.

Upper bounds for the sum of Laplacian eigenvalues of graphs

Linear Algebra and its Applications, 2012

Let G be a graph with n vertices and e(G) edges, and let μ 1 (G) ≥ μ 2 (G) ≥ • • • ≥ μ n (G) = 0 be the Laplacian eigenvalues of G. Let S k (G) = k i=1 μ i (G), where 1 ≤ k ≤ n. Brouwer conjectured that S k (G) ≤ e(G) + k+1 2 for 1 ≤ k ≤ n. It has been shown in Haemers et al. [7] that the conjecture is true for trees. We give upper bounds for S k (G), and in particular, we show that the conjecture is true for unicyclic and bicyclic graphs.

New lower bounds for the energy of matrices and graphs

arXiv: Spectral Theory, 2019

Let RRR be a Hermitian matrix. The energy of RRR, mathcalE(R)\mathcal{E}(R)mathcalE(R), corresponds to the sum of the absolute values of its eigenvalues. In this work it is obtained two lower bounds for mathcalE(R).\mathcal{E}(R).mathcalE(R). The first one generalizes a lower bound obtained by Mc Clellands for the energy of graphs in 197119711971 to the case of Hermitian matrices and graphs with a given nullity. The second one generalizes a lower bound obtained by K. Das, S. A. Mojallal and I. Gutman in 2013 to symmetric non-negative matrices and graphs with a given nullity. The equality cases are discussed. These lower bounds are obtained for graphs with mmm edges and some examples are provided showing that, some obtained bounds are incomparable with the known lower bound for the energy 2sqrtm2\sqrt{m}2sqrtm. Another family of lower bounds are obtained from an increasing sequence of lower bounds for the spectral radius of a graph. The bounds are stated for singular and non-singular graphs.

On conjectures involving second largest signless Laplacian eigenvalue of graphs

Linear Algebra and its Applications, 2010

Let G = (V, E) be a simple graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the Laplacian matrix of G is L(G) = D(G) − A(G) and the signless Laplacian matrix of G is Q (G) = D(G) + A(G). In this paper we obtain a lower bound on the second largest signless Laplacian eigenvalue and an upper bound on the smallest signless Laplacian eigenvalue of G. In [5], Cvetković et al. have given a series of 30 conjectures on Laplacian eigenvalues and signless Laplacian eigenvalues of G (see also [1]). Here we prove five conjectures.

Proof of conjectures involving the largest and the smallest signless Laplacian eigenvalues of graphs

Discrete Mathematics, 2012

Let G = (V , E) be a simple graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the signless Laplacian matrix of G is Q (G) = D(G) + A(G). In [5], Cvetković et al. (2007) have given conjectures on signless Laplacian eigenvalues of G (see also Aouchiche and Hansen (2010) [1], Oliveira et al. (2010) [14]). Here we prove two conjectures.

Inequalities on Vertex Degrees , Eigenvalues and ( Signless ) Laplacian Eigenvalues of Graphs

2010

Several inequalities on vertex degrees, eigenvalues, Laplacian eigen-values, and signless Laplacian eigenvalues of graphs are presented in this note. Some of them are generalizations of the inequalities in [2]. We consider only finite undirected graphs without loops or multiple edges. Notation and terminology not defined here follow that in [1]. We use [n] to denote the set of { 1, 2, ..., n}. For each subset I of [n], we use I c to denote [n] − I. Let G be a graph of order n. We assume that the vertices in G are ordered such that d 1 ≥ d 2 ≥ ... ≥ d n , where d i , 1 ≤ i ≤ n, is the degree of vertex v i in G. We define Σ k (G) as Σ n i=1 d k i. The eigenvalues μ 1 (G) ≥ μ 2 (G) ≥ ... ≥ μ n (G) of the adjacency matrix A(G) of G are called the eigenvalues of the graph G. Let D(G) be the diagonal matrix of the degree sequence of G. The eigenvalues λ 1 (G) ≥ λ 2 (G) ≥ ... ≥ λ n−1 ≥ λ n (G) = 0 of L(G) := D(G)−A(G) of G are called the Laplacian eigenvalues of the graph G. The eigenvalue...