2024 Bounds on eigenvalues and chromatic numbers

Bounds on eigenvalues and chromatic numbers

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August undefined, 2024

WebOct 29, 2012 · New spectral bounds on the chromatic number Clive Elphick, Pawel Wocjan One of the best known results in spectral graph theory is the following lower bound on the chromatic number due to Alan Hoffman, where mu_1 and mu_n are respectively the maximum and minimum eigenvalues of the adjacency matrix: q >= 1 + mu_1 / -mu_n. WebLet 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 ...

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WebApr 7, 2009 · Bounds are obtained for characteristic numbers of graphs, such as the size of a maximal (co)clique, the chromatic number, the diameter, and the bandwidth, in terms of the eigenvalues of the standard adjacency matrix or the Laplacian matrix. We also deal with inequalities and regularity results concerning the structure of graphs and block designs. WebFeb 1, 1979 · It provides some new observations on the Lovász θ-function of regular graphs, calculation of the Shannon capacity of some strongly regular graphs, bounds on eigenvalues of graphs (in... coby bluetooth transmitter

Observations on the Lovasz´ θ-Function, Graph Capacity, …

WebVolume 334. November, 2014. Read More. Publisher: Elsevier Science Publishers B. V. PO Box 211 1000 AE Amsterdam Webeigenvalue. This corresponds to the largest eigenvalue of the Laplacian, which we will examine as well. We will relate these to bounds on the chromatic numbers of graphs and the sizes of independent sets of vertices in graphs. In particular, we will prove Ho man’s bound, and some generalizations. Webchromatic number, denoted by ˜ k(G), which is just the chromatic number of Gk. Hence, ˜ k(G) = ˜(Gk). It is well known that 1(G) = (G) n=˜(G). Therefore, lower bounds on the k-distance chromatic number can be obtained by nding upper bounds on the corre-sponding k-independence number, and vice versa. The parameter k has also been studied calling the uk from us cell phone

Nordhaus–Gaddum and other bounds for the sum of

Spectral bounds for the independence ratio and the chromatic …

Webk. This article proves various eigenvalue bounds for the independence number and chromatic number of Gk which purely depend on the spectrum of G, together with a … WebBounds on Eigenvalues and Chromatic Numbers Dasong Cao School of Industrial and System Engineering Georgia Institute of Technology Atlanta, Georgia 30332 Submitted by Richard A. Brualdi ABSTRACT We give new bounds on eigenvalue of graphs which imply some known bounds. In particular, if T(G) is the ... calling the uk from uaeWebSep 28, 2024 · Theory Ser. B.97 (2007) 859–865) conjectured the following. If G is a Kr+1 -free graph on at least r+ 1 vertices and m edges, then , where λ1 ( G )and λ2 ( G) are the … calling the united kingdom from usa

"WebThis gives a lower bound on the chromatic number of 4:2, which implies a lower bound of 5. We can improve the lower bound by re-weighting the edges of the graph. For example, if we give weight 2 to all the edges in the clique and weight 1 to all the others, we obtain a bound of 5:18, which agrees with the chromatic number of this graph which is 6. " - Bounds on eigenvalues and chromatic numbers

Bounds on eigenvalues and chromatic numbers

WebJan 1, 2024 · In the present paper we are interested in the study of the distance Laplacian eigenvalues of a connected graph with fixed order n and chromatic number χ. We … WebCao D., Bounds on eigenvalues and chromatic numbers, Linear Algebra and its Applications, 1998, 270 :1–13 MATH MathSciNet Google Scholar Cvetkovic D. M., Doob M., Sachs H., Spectra of Graphs, New York: Academic Press, 1980 Brualdi R. A., Hoffman A. J., On the spectral radius of 0-1 matrices, Linear Algebra and its Applications, 1985, 65 …

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WebJan 15, 2007 · Cao, Bounds on eigenvalues and chromatic numbers, Linear Algebra Appl. 270 (1998) 1–13. [3] D. Cvetkovi´c, M. Doob, H. Sachs, Spectra of Graphs, VEB Deutscher Verlag der Wissenschaften, Berlin, 1980, 368pp. [4] K. Das, P. Kumar, Some new bounds on the spectral radius of graphs, Discrete Math. 281 (2004) 149–161. [5] O. http://www-personal.umich.edu/~mmustata/Slides_Lecture13_565.pdf

WebJun 20, 2014 · We define the independence ratio and the chromatic number for bounded, self-adjoint operators on an L 2-space by extending the definitions for the adjacency … WebThis dissertation involves combining the two concepts of energy and the chromatic number of classes of graphs into a new ratio, the eigen-chromatic ratio of a graph G. Associated with this ratio is the importance of its asymptotic convergence in

WebThe generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E … Web(9) Lower bounds on the chromatic number of strong products of graphs are expressed in terms of the order and Lova´sz θ-function of each factor (Proposition 4). Their utility is exempliﬁed, while also leading to exact chromatic numbers in some cases. The paper is structured as follows: Section II provides notation and a thorough review of

Webboundsfor the k-independence numberand k-chromatic number, together with a method to optimize them. In particular, such bounds are shown to be tight for some of the so-called …

WebBounds on the chromatic number Last class, I introduced proper colorings of graphs, and the chromatic number. We also looked at some bounds on the chromatic number, … coby bluetooth speaker not workingWebCapacity, Eigenvalues, and Strong Products Igal Sason Dedicated to my friend and former teacher, Professor Emeritus Abraham (Avi) Berman, in the occasion of his eightieth birthday Citation: I. Sason, “Observations on the Lova´sz θ … coby boschWebeigenvalue. This corresponds to the largest eigenvalue of the Laplacian, which we will examine as well. We will relate these to bounds on the chromatic numbers of graphs and the sizes of independent sets of vertices in graphs. In particular, we will prove Ho↵man’s bound, and some generalizations. coby boulware baseballWebThis is the first known eigenvalue bound for the max-k-cut when k>2 that is applicable to any graph. This bound is exploited to derive a new eigenvalue bound on the chromatic number of a graph. For regular graphs, the new bound on the chromatic number is the same as the well-known Hoffman bound; however, the two bounds are incomparable in … coby bluetooth speaker pairingWebThe organizers of the Algebraic Graph Theory International Webinar would like to invite you to join us and other colleagues on March 21, 2024, at 7pm Central European Summer Time (= 6pm UTC), for the next presentation delivered by Veronika Bachrata.. The title: Eigenvalue bounds for the independence and chromatic number of graph powers … calling the usa from englandWebOct 1, 2024 · Bounds for s + Similarly we can consider upper and lower bounds for s + ( G) + s + ( G ‾). First, we prove a lower bound. Theorem 4 For any graph G: s + ( G) + s + ( G ‾) > ( n − 1) 2 2. Proof Using the well-known inequality μ ( G) ≥ 2 m / n we get: s + ( G) + s + ( G ‾) ≥ 4 m 2 n 2 + ( n ( n − 1) − 2 m) 2 n 2. coby bouwmansWebJun 17, 2016 · Abstract. In [3] A. J. Hoffman proved a lower bound on the chromatic number of a graph in the terms of the largest and the smallest eigenvalues of its adjacency matrix. In this paper, we prove a higher dimensional version of this result and give a lower bound on the chromatic number of a pure d -dimensional simplicial complex in the … coby boat