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Spectral bounds for the clique and independence numbers of graphs

1986
*
Journal of combinatorial theory. Series B (Print)
*

We give lower

doi:10.1016/0095-8956(86)90069-9
fatcat:d33pz3w4h5ekxcikh2mcq3r7sm
*bounds**for**the**clique**number*k(G)*and**for**the*(vertex)*independence**number*a(G)*of*a*graph*G. They improve earlier results,*and*they involve*the*spectrum*of**the**graph*G. ... . < k,(G)*of*lower*bounds**for**the**clique**number*(size*of**the*largest*clique*)*of*a*graph*G*of*n vertices.*The**bounds*involve*the*spectrum*of**the*adjacency matrix*of*G. ...##
###
A generalization for the clique and independence numbers

2012
*
The Electronic Journal of Linear Algebra
*

In this paper, lower

doi:10.13001/1081-3810.1512
fatcat:m4xxkxbdoja7lhdogoj7se7jty
*and*upper*bounds**for**the**clique**and**independence**numbers*are established in terms*of**the*eigenvalues*of**the*signless Laplacian matrix*of*a given*graph*G. http://math.technion.ac.il ... matrix*of*a*graph**and*obtained a lower*and*upper*bound*on*the**independence**and**clique**numbers**for*regular*graphs*. ... We note ELA A Generalization*for**the**Clique**and**Independence**Numbers*165 that, in*the*literature, several studies have been presented about*the*connections between*the**clique**number**of*G*and**the**spectral*...##
###
Bypassing the XOR Trick: Stronger Certificates for Hypergraph Clique Number
[article]

2022
*
arXiv
*
pre-print

*For*example,

*for*any constant p, with high probability over

*the*choice

*of*

*the*hypergraph, our

*spectral*algorithm certifies a

*bound*

*of*Õ(√(n)) on

*the*

*clique*

*number*in polynomial time. ... This matches, up to polylog(n) factors,

*the*best known certificate

*for*

*the*

*clique*

*number*in random

*graphs*, which is

*the*special case

*of*k = 2. ... upper

*bounds*on

*the*

*clique*

*number*

*of*random

*graphs*. ...

##
###
Spectral radius and clique partitions of graphs

2021
*
Linear Algebra and its Applications
*

We give lower

doi:10.1016/j.laa.2021.07.025
fatcat:orixonkvwbd4rh3hc2zucxgko4
*bounds*on*the*size*and*total size*of**clique*partitions*of*a*graph*in terms*of*its*spectral*radius*and*minimum degree,*and*derive a*spectral*upper*bound**for**the*maximum*number**of*edge-disjoint ...*The*extremal*graphs*attaining*the**bounds*are exactly*the*block*graphs**of*Steiner 2-designs*and**the*regular*graphs*with K_t-decompositions, respectively. ... This work is supported by*the*National Natural Science Foundation*of*China (No. 11801115*and*No. 12071097)*and**the*Fundamental Research Funds*for**the*Central Universities. ...##
###
New Bounds on the Clique Number of Graphs Based on Spectral Hypergraph Theory
[chapter]

2009
*
Lecture Notes in Computer Science
*

This work introduces new

doi:10.1007/978-3-642-11169-3_4
fatcat:isf63cbxxzhajh7lulelgjiec4
*bounds*on*the**clique**number**of**graphs*derived from a result due to Sós*and*Straus, which generalizes*the*Motzkin-Straus Theorem to a specific class*of*hypergraphs. ... In particular, we generalize*and*improve*the**spectral**bounds*introduced by Wilf in 1967*and*1986 establishing an interesting link between*the**clique**number**and**the*emerging*spectral*hypergraph theory field ... In 1967, Wilf [4] used*for**the*first time*spectral**graph*theory*for*computing*bounds*on*the**clique**number**of**graphs*. ...##
###
GRAPHS WITH SMALL INDEPENDENCE NUMBER MINIMIZING THE SPECTRAL RADIUS

2013
*
Discrete Mathematics, Algorithms and Applications (DMAA)
*

*The*

*independence*

*number*

*of*a

*graph*is defined as

*the*maximum size

*of*a set

*of*pairwise non-adjacent vertices

*and*

*the*

*spectral*radius is defined as

*the*maximum eigenvalue

*of*

*the*adjacency matrix

*of*

*the*... Xu et al. in [

*The*minimum

*spectral*radius

*of*

*graphs*with a given

*independence*

*number*, Linear Algebra

*and*its Applications 431 (2009) 937-945] determined

*the*connected

*graphs*

*of*order n with

*independence*...

*The*Laplacian

*spectral*

*bounds*

*for*

*clique*

*and*

*independence*

*numbers*

*of*

*graphs*have been shown in [8, 13] . ...

##
###
The bounds of the spectral radius of general hypergraphs in terms of clique number
[article]

2020
*
arXiv
*
pre-print

As an application, we finally obtain an upper

arXiv:2007.13282v1
fatcat:u5pt7vxpxjhcxcruwpe33mwj6e
*bound**of**the**spectral*radius*of*general hypergraphs in terms*of**clique**number*. ... In this paper, we firstly obtain a lower*bound**of**the**spectral*radius (or*the*signless Laplacian*spectral*radius)*of*general hypergraphs in terms*of**clique**number*. ... [16] presented some lower*and*upper*bounds**for**the**independence**number**and**the**clique**number*involving*the*Laplacian eigenvalues*of*a*graph*G. ...##
###
uran's theorem implies Stanley's bound

2020
*
Discussiones Mathematicae Graph Theory
*

Moreover, writing ω

doi:10.7151/dmgt.2287
fatcat:5fxzoxpasvdnzgd4olwmvaih4i
*for**the**clique**number**of*G*and*W k*for**the**number**of*its walks on k vertices, it is shown that*the*sequence is nonincreasing*and*converges to ρ. ... Let G be a*graph*with m edges*and*let ρ be*the*largest eigenvalue*of*its adjacency matrix. It is shown that improving*the*well-known*bound**of*Stanley. ... Indeed, let G be a*graph*with m edges,*clique**number*ω,*and**spectral*radius ρ. Write W k*for**the**number**of*walks on k vertices in G. ...##
###
Hardness results and spectral techniques for combinatorial problems on circulant graphs

1998
*
Linear Algebra and its Applications
*

We show that computing (

doi:10.1016/s0024-3795(98)10126-x
fatcat:i375n76sm5dthoaytuwuhp4usi
*and*even approximating) MAXIMUM*CLIQUE**and*MINI- MUM*GRAPH*COLORING*for*circulant*graphs*is essentially as hard as in*the*general case. ... ., prime order*and*/or sparseness, GRAPHISOMORPHISM*and*MINIMUMGRAPHCOLORING hecomeeasierin*the*circulant case,*and*we take advantage*of**spectral*techniques*for*their efficient computation. 0 1998 Elsevier ... Acknowledgements We thank Davide Ferrario*for**the*useful topological insights that led to a great simplification*of**the*proof*of*Theorem 9. ...##
###
Spectral techniques in graph algorithms
[chapter]

1998
*
Lecture Notes in Computer Science
*

Introduction

doi:10.1007/bfb0054322
fatcat:fdonninotffvdb3mj5jrqgk4li
*Graph*bisection,*graph*coloring*and*finding*the**independence**number**of*a*graph*are three well studied algorithmic problems. ...*The*existence*of*efficient algorithms to compute*the*eigenvectors*and*eigenvalues*of**graphs*supplies a useful tool*for**the*design*of*various*graph*algorithms. ...*The*relevance*of**graph*eigenvalues to*cliques*or*independent*sets in*the**graphs*is well known*and*can be traced back to*the*old result that*the**independence**number**of*any regular*graph*is at most −nλ n ...##
###
More spectral bounds on the clique and independence numbers
[article]

2017
*
arXiv
*
pre-print

We give some new

arXiv:0706.0548v2
fatcat:foeh2ibmdbhztbgl7hm7vhncga
*bounds**for**the**clique**and**independence**numbers**of*a*graph*in terms*of*its eigenvalues. ... Given a*graph*G, we write ω (G)*and*α (G)*for*its*clique**and**independence**numbers*. ... Introduction*and*main results In this note we give some new relations between*graph*spectra*and**clique**and**independence**numbers*, a topic recently studied in [8] , [9]*and*[12] . ...##
###
Conjectured lower bound for the clique number of a graph
[article]

2018
*
arXiv
*
pre-print

It is well known that n/(n - μ), where μ is

arXiv:1804.03752v2
fatcat:6yt7piittbaepfnbzbfbq3ff5i
*the**spectral*radius*of*a*graph*with n vertices, is a lower*bound**for**the**clique**number*. ... We prove this conjecture*for*various classes*of**graphs*, including triangle-free*graphs*,*and**for*almost all*graphs*. ... Acknowledgements This research was supported in part by*the*National Science Foundation Award 1525943 . ...##
###
Coloring the normalized Laplacian for oriented hypergraphs
[article]

2021
*
arXiv
*
pre-print

*For*

*the*

*independence*

*number*, both an inertia--like

*bound*

*and*a ratio--like

*bound*are shown. ... A Sandwich Theorem involving

*the*

*clique*

*number*,

*the*vector chromatic

*number*

*and*

*the*coloring

*number*is proved, as well as a lower

*bound*

*for*

*the*vector chromatic

*number*in terms

*of*

*the*smallest

*and*

*the*... Acknowledgments

*The*authors are grateful to

*the*anonymous referee

*for*

*the*comments

*and*suggestions that have greatly improved

*the*first version

*of*this paper.

*The*research

*of*A. ...

##
###
On spectral spread of generalized distance matrix of a graph
[article]

2019
*
arXiv
*
pre-print

We also obtain lower

arXiv:1907.09462v1
fatcat:btbhz4wwizbrzasurtc7k3rq7m
*bounds**for*D_αS(G) in terms*of**clique**number**and**independence**number**of**the**graph*G*and*characterize*the*extremal*graphs**for*some cases. ... Further, we obtain*the*lower*bounds**for*D_αS(G)*of*bipartite*graphs*involving different*graph*parameters*and*we characterize*the*extremal*graphs**for*some cases. ... In Section 4, we obtain lower*bounds**for**the*generalized distance*spectral*spread D α (G) in terms*of**clique**number**and**independence**number**of**the**graph*G*and*characterize*the*extremal*graphs**for*some ...##
###
New measures of graph irregularity

English

2014
*
Electronic Journal of Graph Theory and Applications
*

English

In this paper, we tighten

doi:10.5614/ejgta.2014.2.1.5
fatcat:6rxtq5fh2fedfdisnvxc5jhygy
*the*concise Turán theorem*for*irregular*graphs*, using*spectral**and*non-*spectral*proofs. ... We then investigate to what extent Turán's theorem can be similarly strengthened*for*generalized r-partite*graphs*. ... This work was supported in part by*the*National Science Foundation Science*and*Technology Centre*for*Science*of*Information, under grant CCF-0939370. ...
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