Conjectured lower bound for the clique number of a graph
aa r X i v : . [ m a t h . C O ] A ug Conjectured lower bound for the clique number of agraph
Clive Elphick ∗ Pawel Wocjan † April 10, 2018
Abstract
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 conjecture that µ can be replacedin this bound with √ s + , where s + is the sum of the squares of the positive eigenvalues.We prove this conjecture for various classes of graphs, including triangle-free graphs,and for almost all graphs. Let G be a graph, with no isolated vertices, with n vertices, edge set E with | E | = m ,average degree d , chromatic number χ ( G ) and clique number ω ( G ). We also let A denote the adjacency matrix of G and let µ = µ ≥ . . . ≥ µ n denote the eigenvalues of A . The inertia of A is the ordered triple ( π, ν, γ ) where π , ν and γ are the numberscounting muliplicities of positive, negative and zero eigenvalues of A respectively. Let s + = π X i =1 µ i and s − = n X i = n − ν +1 µ i . Note that: n X i =1 µ i = tr( A ) = 2 m = s + + s − . µ with s + Edwards and Elphick [5] proved that 2 m m − µ ≤ χ ( G ) ∗ [email protected], School of Mathematics, University of Birmingham, Birmingham, UK † [email protected], Department of Computer Science, University of Central Florida, Florida, USA nd Ando and Lin [1] proved a conjecture due to Wocjan and Elphick [15] that2 m m − s + = 1 + s + s − ≤ χ ( G ) . As another example of replacing µ with s + , Hong [9] proved for graphs with noisolated vertices that µ ≤ m − n + 1, and Elphick et al [6] proved that for almost allconnected graphs s + ≤ m − n +1. Similarly Favaron et al [7] proved that ω ( G ) ≤ m/µ and Wu and Elphick [16] proved the doubly stronger result that χ ( G ) ≤ m/ √ s + . Finally Stanley [12] proved that µ ≤ √ m + 1 − , and Wu and Elphick [16] proved that √ s + ≤ √ m + 1 − . So in all of these cases we can strengthen known bounds by replacing µ with s + .The next section considers the same replacement for a well known lower bound for theclique number. The concise version of Tur´an’s theorem states that: nn − d ≤ ω ( G ) . (1)This bound was improved by Caro [4] and Wei [13] using degrees as follows: n X i =1 n − d i ≤ ω ( G );and by Wilf [14] using the spectral radius as follows: nn − µ ≤ ω ( G ) . (2)Bound (2) was strengthened by Nikiforov [11] who proved the following conjectureof Edwards and Elphick [5]. 2 m m − µ ≤ ω ( G ) . (3)Note that for regular graphs, all of these bounds equal n/ ( n − d ). Wocjan andElphick [15] noted that 2 m m − s + ω ( G ) . n alternative strengthening of Wilf’s bound is provided by the following conjec-ture, which we have tested against the thousands of named graphs with up to 40 verticesin the Wolfram Mathematica database, and found no counter-example. Aouchiche [2]has tested this conjecture using his powerful AGX software, and also found no counter-example. Conjecture 1 exceeds n/ ( n − d ) for all regular graphs with more than onepositive eigenvalue. Conjecture 1.
For any graph
G nn − √ s + ≤ ω ( G ) . This conjecture is exact, for example, for complete regular multipartite graphs.
We can prove this conjecture for the following classes of graphs.
Proof.
Let t denote the number of triangles in a graph. It is well known that: n X i =1 µ i = tr( A ) = 6 t, so for triangle-free graphs π X i =1 µ i = − n X i = n − ν +1 µ i . Therefore, using that µ ≥ | µ n | s − ≥ P ni = n − ν +1 µ i µ n = P πi =1 µ i | µ n | ≥ µ | µ n | ≥ µ . Therefore, using the lower bound on the largest eigenvalue µ ≥ m/n , the equality ( s + + s − ) = m combined with the arithmetic-geometric-mean inequality, and theabove lower bound on s − , we obtain √ s + ≤ µ n m √ s + ≤ n m √ s − √ s + ≤ n m m n . Proof.
Weakly perfect graphs have ω ( G ) = χ ( G ). Therefore using the result due toAndo and Lin [1] discussed above and that µ ≥ m/n : nn − √ s + ≤ m m − s + ≤ χ ( G ) = ω ( G ) . .3 Proof for some strongly regular graphs We do not know how to prove this conjecture for all strongly regular graphs. How-ever we can prove the conjecture for the subset of strongly regular graphs which areKneser graphs. The Kneser graph KG p,k is the graph whose vertices correspond tothe k − element subset of a set of p elements, and where two vertices are joined if andonly if the corresponding sets are disjoint. The Kneser graphs with k = 2 are stronglyregular, with only three distinct eigenvalues. For these graphs n = (cid:18) p (cid:19) , ω = j p k , m = (cid:18) p (cid:19)(cid:18) p − (cid:19) and p ≥ k = 4 . The eigenvalues (see Godsil and Royle [8]) are:( − i (cid:18) p − − i − i (cid:19) with multiplicity (cid:18) pi (cid:19) − (cid:18) pi − (cid:19) , for i = 0 , , . We are seeking to prove that nn − √ s + ≤ p − ≤ j p k = ω ( KG p, ) , which re-arranges to s + = 2 m − s − ≤ n ( p − ( p − = p ( p − . Inserting the negative eigenvalues this becomes (cid:18) p (cid:19)(cid:18) p − (cid:19) − ( p − (cid:18) p − (cid:19) ≤ p ( p − . Simple algebra reduces this to 2 p − p + 6 ≥ p ≥ Proof.
We use the Erdos-Renyi random graph model G n ( p ), which consists of all graphswith n vertices in which edges are chosen independently with probability p . Bollob´asand Erdos [3] proved that the clique number is almost always x or x + 1 where x = 2 log n log(1 /p ) + O (log log n ) . Since almost all graphs have all degrees very close to n/ p = 0 .
5. Therefore s + ≤ s + + s − = 2 m ≈ n n n − √ s + ≤ nn − n/ √ ≈ . < n log 2 ≈ ω ( G ) . Lower bounds for the clique number are often proved using the Motzkin-Straus [10]inequality, which can be expressed as follows. For any adjacent vertices i and j suchthat i < j we write i ∼ j . Then for any vector ( p , . . . , p n ) with p i ≥ i and P ni =1 p i = 1: X i ∼ j p i p j ≤ ω − ω . It is however not evident how to use this approach in the context of Conjecture 1,where the number of positive eigenvalues varies greatly between graphs with n vertices. Acknowledgements
This research was supported in part by the National Science Foundation Award 1525943.
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