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Spectral bounds for the clique and independence numbers of graphs
1986
Journal of combinatorial theory. Series B (Print)
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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
2012
The Electronic Journal of Linear Algebra
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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]
2022
arXiv
pre-print
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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
2021
Linear Algebra and its Applications
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2021 pre-print arXiv:2111.02734v1 fatcat:a5oqllbpijd7bkr4p26w3r3qva
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]
2009
Lecture Notes in Computer Science
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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
2013
Discrete Mathematics, Algorithms and Applications (DMAA)
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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]
2020
arXiv
pre-print
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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
2020
Discussiones Mathematicae Graph Theory
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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
1998
Linear Algebra and its Applications
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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]
1998
Lecture Notes in Computer Science
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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]
2017
arXiv
pre-print
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2007 pre-print arXiv:0706.0548v1 fatcat:6zc62geurzbffd3fty4ddwklsu
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]
2018
arXiv
pre-print
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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]
2021
arXiv
pre-print
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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]
2019
arXiv
pre-print
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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
2014
Electronic Journal of Graph Theory and Applications
English
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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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