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Spectral bounds for the clique and independence numbers of graphs

Herbert S Wilf
1986 Journal of combinatorial theory. Series B (Print)  
We give lower 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.  ... 
doi:10.1016/0095-8956(86)90069-9 fatcat:d33pz3w4h5ekxcikh2mcq3r7sm

A generalization for the clique and independence numbers

A. Dilek Maden, A. Sinan Cevik
2012 The Electronic Journal of Linear Algebra  
In this paper, lower 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  ... 
doi:10.13001/1081-3810.1512 fatcat:m4xxkxbdoja7lhdogoj7se7jty

Bypassing the XOR Trick: Stronger Certificates for Hypergraph Clique Number [article]

Venkatesan Guruswami, Pravesh K. Kothari, Peter Manohar
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.  ... 
arXiv:2205.06739v1 fatcat:n5pjihzwv5d4rfmxk7g32kccdy

Spectral radius and clique partitions of graphs

Jiang Zhou, Edwin R. van Dam
2021 Linear Algebra and its Applications  
We give lower 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.  ... 
doi:10.1016/j.laa.2021.07.025 fatcat:orixonkvwbd4rh3hc2zucxgko4

New Bounds on the Clique Number of Graphs Based on Spectral Hypergraph Theory [chapter]

Samuel Rota Bulò, Marcello Pelillo
2009 Lecture Notes in Computer Science  
This work introduces new 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.  ... 
doi:10.1007/978-3-642-11169-3_4 fatcat:isf63cbxxzhajh7lulelgjiec4

GRAPHS WITH SMALL INDEPENDENCE NUMBER MINIMIZING THE SPECTRAL RADIUS

XUE DU, LINGSHENG SHI
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] .  ... 
doi:10.1142/s1793830913500171 fatcat:736za2w6pfbdzcc2wrta7eqzsm

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

Cunxiang Duan, Ligong Wang
2020 arXiv   pre-print
As an application, we finally obtain an upper 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.  ... 
arXiv:2007.13282v1 fatcat:u5pt7vxpxjhcxcruwpe33mwj6e

uran's theorem implies Stanley's bound

Vladimir Nikiforov
2020 Discussiones Mathematicae Graph Theory  
Moreover, writing ω 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.  ... 
doi:10.7151/dmgt.2287 fatcat:5fxzoxpasvdnzgd4olwmvaih4i

Hardness results and spectral techniques for combinatorial problems on circulant graphs

Bruno Codenotti, Ivan Gerace, Sebastiano Vigna
1998 Linear Algebra and its Applications  
We show that computing (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.  ... 
doi:10.1016/s0024-3795(98)10126-x fatcat:i375n76sm5dthoaytuwuhp4usi

Spectral techniques in graph algorithms [chapter]

Noga Alon
1998 Lecture Notes in Computer Science  
Introduction 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  ... 
doi:10.1007/bfb0054322 fatcat:fdonninotffvdb3mj5jrqgk4li

More spectral bounds on the clique and independence numbers [article]

Vladimir Nikiforov
2017 arXiv   pre-print
We give some new 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] .  ... 
arXiv:0706.0548v2 fatcat:foeh2ibmdbhztbgl7hm7vhncga

Conjectured lower bound for the clique number of a graph [article]

Clive Elphick, Pawel Wocjan
2018 arXiv   pre-print
It is well known that n/(n - μ), where μ is 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 .  ... 
arXiv:1804.03752v2 fatcat:6yt7piittbaepfnbzbfbq3ff5i

Coloring the normalized Laplacian for oriented hypergraphs [article]

Aida Abiad, Raffaella Mulas, Dong Zhang
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.  ... 
arXiv:2008.03269v2 fatcat:43ubktlihrh6fgcizz7naur2pm

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

Hilal A. Ganie, S. Pirzada, A. Alhevaz, M. Baghipur
2019 arXiv   pre-print
We also obtain lower 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  ... 
arXiv:1907.09462v1 fatcat:btbhz4wwizbrzasurtc7k3rq7m

New measures of graph irregularity
English

Clive Elphick, Pawel Wocjan
2014 Electronic Journal of Graph Theory and Applications  
In this paper, we tighten 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.  ... 
doi:10.5614/ejgta.2014.2.1.5 fatcat:6rxtq5fh2fedfdisnvxc5jhygy
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