A generalization for the clique and independence numbers

European Journal of Pure and Applied Mathematics

Core-satellite graphs Θ(c, s, η) ∼= Kc ▽ (ηKs) are graphs consisting of a central clique Kc (the core) and η copies of Ks (the satellites) meeting in a common clique. They belong to the class of graphs of diameter two. Agave graphs Θ(2, 1, η) ∼= K2 ▽ (ηK1) belong to the general class of complete split graphs, where the graphs consist of a central clique K2 and η copies of K1 which are connected to all the nodes of the clique. They are the subclass of Core-satellite graphs. Let μ(G) be the spectral radius of the signless Laplacian matrix Q(G). In this paper, we have obtained the greatest lower bound and the least upper bound of signless Laplacian spectral radius of Agave graphs. These bounds have been expressed in terms of graph invariants like m the number of edges, n the number of vertices, δ the minimum degree, ∆ the maximum degree, and η copies of the satellite. We have made use of the approximation technique to derive these bounds. This unique approach can be utilized to determi...

Discrete Applied Mathematics, 2010

Let G = (V , E) be a simple, undirected graph of order n and size m with vertex set V , edge set E, adjacency matrix A and vertex degrees ∆

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.

Let L n,t be the set of all n-vertex connected graphs with clique number t (2 ≤ t ≤ n). For n-vertex connected graphs with given clique number, lexicographic ordering by spectral moments (S-order) is discussed in this paper. The first

Discrete Applied Mathematics, 2017

For a simple graph G with n-vertices, m edges and having signless Laplacian eigenvalues q 1 , q 2 ,. .. , q n , the signless Laplacian energy QE(G) of the graph G is defined as QE(G) =  n i=1 | q i − d |, where d = 2m n is the average degree of G. In this paper, we obtain the lower and upper bounds for the signless Laplacian energy QE(G) in terms of clique number ω, maximum degree ∆, number of vertices n, first Zagreb index M 1 (G) and number of edges m. As an application, we obtain the bounds for the energy of line graph L (G) of a graph G in terms of various graph parameters. We also obtain a relation between the signless Laplacian energy QE(G) and the incidence energy IE(G).

Linear Algebra and its Applications, 2012

The Laplacian spectral radius of a graph G is the largest eigenvalue of its Laplacian matrix. In this paper, the first three smallest values of the Laplacian spectral radii among all connected graphs with clique number ω are obtained.

2012

One of the best known results in spectral graph theory is the following lower bound on the chromatic number due to Alan Hoffman, where µ 1 and µ n are respectively the maximum and minimum eigenvalues of the adjacency matrix: χ ≥ 1 + µ 1 / − µ n. Liliya Kolotilina has demonstrated that −µ n in this bound can be replaced with either of two formulae, which involve maximum and minimum eigenvalues of the adjacency, Laplacian and signless Laplacian matrices. In this paper we use majorization to generalize Kolotilina's bounds, by encompassing all eigenvalues of all three matrices. We compare the performance of these bounds for named and random graphs and derive several known bounds as corollaries of Kolotilina's bounds. Our proof relies on a technique of converting the adjacency matrix into the zero matrix by conjugating with χ diagonal unitary matrices. We also prove that the minimum number of diagonal unitary matrices that can be used to convert the adjacency matrix into the zero matrix is the normalized orthogonal rank, which can be strictly less than the chromatic number.

Discrete Mathematics, 2019

This paper generalizes and unifies the existing spectral bounds on the k-independence number of a graph, which is the maximum size of a set of vertices at pairwise distance greater than k. The previous bounds known in the literature follow as a corollary of the main results in this work. We show that for most cases our bounds outperform the previous known bounds. Some infinite graphs where the bounds are tight are also presented. Finally, as a byproduct, we derive some lower spectral bounds for the diameter of a graph.

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.

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...