AA graph for which the inertia bound is not tight
John Sinkovic ∗ October 19, 2018
Abstract
The inertia bound gives an upper bound on the independence number of a graphby considering the inertia of matrices corresponding to the graph. The bound is knownto be tight for graphs on 10 or fewer vertices as well as for all perfect graphs. Thequestion has been asked as to whether the bound is always tight. We show that thebound is not tight for the Paley graph on 17 vertices as well as for the graph obtainedfrom Paley 17 by deleting a vertex.
AMS 2010 subject classification:
Keywords: inertia bound, Cvetkovi´c bound, independence number, weight matrix
In algebraic graph theory it is common to define a matrix or family of matrices using a simplegraph. Properties of a such a matrix or matrices are then analyzed to make connections to aproperty of the graph. One such connection exists between the independence number α ( G )and the eigenvalues of weight matrices W corresponding to G . A weight matrix W of G isa real symmetric matrix such that w ij = 0 if ij is not an edge of G . For any weight matrix W of G , α ( G ) ≤ min {| G | − n + ( W ) , | G | − n − ( W ) } . (See Lemma 9.6.3 in [4].) This upper bound on α is attributed to D.M. Cvetkovi´c (PhDthesis 1971). It is often referred to as the Cvetkovi´c bound or inertia bound.As with many inequalities, it is interesting to determine when equality occurs or in otherwords when the bound is tight. The following question was posed by Chris Godsil andappears in [2, 3, 5]. ∗ Department of Combinatorics and Optimization, University of Waterloo, ( [email protected] ) Interesting Graphs and their Colourings , unpublished lecture notes C. Godsil (2004) a r X i v : . [ m a t h . C O ] S e p uestion 1. Does each graph G have a weight matrix W such that α ( G ) = min {| G | − n + ( W ) , | G | − n − ( W ) } ? The inertia bound has been shown to be tight for small graphs [2, 3] (all graphs on 10or fewer vertices, vertex-transitive graphs on 12 or fewer vertices), certain families of Cayleyand strongly regular graphs [5], and perfect graphs . The smallest vertex-transitive graphfor which the tightness of the bound has not been determined is Paley 13 ([2, 3, 5]). Paley17 is the first graph for which a proof has been given showing that the inertia bound is nottight. Let G = ( V, E ) be a simple graph of order n with vertex set V and edge set E . A weightmatrix W of G is an n × n real, symmetric matrix such that w ij = 0 if ( i, j ) (cid:54)∈ E ( G ). As G is simple, it has no loops, and w ii = 0 for all i . In example, any multiple of the adjacencymatrix of G is a weight graph for G . Let n + ( W ) , n − ( W ), and n ( W ) denote the number ofpositive eigenvalues, the number of negative eigenvalues, and the multiplicity of zero as aneigenvalue of W , respectively.Given any weight matrix W of G , the Cvetkovi´c bound or inertia bound states that α ( G ) ≤ min { n − n + ( W ) , n − n − ( W ) } . (1)The following is an equivalent form of the inertia bound which will be useful to our discussion.For any weight matrix W of G , α ( G ) ≤ n ( W ) + min { n + ( W ) , n − ( W ) } . (2)As our goal is to exhibit a graph G for which the bound is not tight, we point out thatin Equation (2) equality cannot occur when both n + ( W ) and n − ( W ) are greater than α ( G ).The claim is that all weight matrices for Paley 17 have at least 4 positive and 4 negativeeigenvalues while the independence number is 3. We prove the claim by looking at invertibleprincipal submatrices corresponding to induced subgraphs of order 7. Interlacing plays a bigpart in our proof and as such we include this well-known theorem. Theorem 2. (Theorem 9.1.1 of [4]) Let A be an n × n real symmetric matrix with eigenvalues λ ≥ λ ≥ . . . ≥ λ n and let C be a k × k principal submatrix of A with eigenvalues τ ≥ τ ≥ . . . ≥ τ k . Then λ i ≥ τ i for all i ∈ { , . . . , k } . A weight matrix W of G is optimal if α ( G ) = min {| G | − n + ( W ) , | G | − n − ( W ) } . Lemma 3.
Let W be a weight matrix for a graph G of order n . If W has a principalsubmatrix with α ( G )+1 positive eigenvalues and a principal submatrix with α ( G )+1 negativeeigenvalues, then W is not optimal. Interesting Graphs and their Colourings , unpublished lecture notes C. Godsil (2004) roof. By Theorem 2, W has at least α ( G ) + 1 positive and α ( G ) + 1 negative eigenvalues.So n − ( W ) ≤ n − α ( G ) − n + ( W ) ≤ n − α ( G ) −
1. Thus min { n − n + ( W ) , n − n − ( W ) } ≥ α ( G ) + 1 > α ( G ). Therefore Equation (1) is not tight and W is not optimal.The independence number of Paley 17 is 3 (Observation 4). We will use Lemma 3 toshow that the inertia bound is not tight for Paley 17. To do so, we need to show that everyweight matrix for Paley 17 has principal submatrices which contain 4 positive eigenvaluesand 4 negative eigenvalues. We now pause for a moment to consider properties of Paley 17. Paley graphs were firststudied in connection with number theory and have been used in giving lower bounds fordiagonal Ramsey numbers. The following definition is taken from [4]. Henceforth q is aprime power such that q ≡ Paley graph P ( q ) has as vertex set the elementsof the finite field GF ( q ), with two vertices adjacent if and only if their difference is a nonzerosquare in GF ( q ).Paley graphs are (among other things) strongly regular graphs, self-complementary, cir-culants, vertex-transitive, edge transitive, and arc-transitive (See [1] chapter XIII).An independent set or coclique is a subset of vertices which are pairwise nonadjacent.The size of a maximum independent set is the independence number of G and is denoted α ( G ). The independence number for a large number of Paley graphs has been calculated byJames B. Shearer . Observation 4.
The independence number of P (17) is 3. A graph G is α -critical if α ( G ) < α ( G − e ) for all edges e . Observation 5.
The graph P (17) is α -critical.Proof. Since P (17) is arc-transitive it is sufficient to show that deleting edge (0 ,
1) increases α . The set { , , , } is independent in P (17) delete (0 , automorphism of a graph G is a permutation σ of V ( G ) such that v ∼ v if and onlyif σ ( v ) ∼ σ ( v ). Lemma 6.
The function σ ab : V → V , σ ab ( v ) = av + b where a is a nonzero square in GF ( q ) and b ∈ GF ( q ) is an automorphism of P ( q ) . Independence number of Paley graphs–IBM Research
Proof.
Note that σ ab is one-to-one and onto. Given two vertices v , v of P ( q ), a nonzerosquare a in GF ( q ), and their images σ ab ( v ) , σ ab ( v ). Now v ∼ v if and only if v − v is a nonzero square. Since a is a nonzero square, v − v is a nonzero square if and onlyif a ( v − v ) is a nonzero square. Further, a ( v − v ) is a nonzero square if and only if av − b − ( av − b ) is a nonzero square if and only if σ ab ( v ) ∼ σ ab ( v ).The quadratic residues (or nonzero squares) modulo 17 are ± , ± , ± , ±
8. A k - edge of P (17) is an edge v v such that v − v = ± k (mod 17) for k ∈ { , , , } . For a given k , theset of k edges form a 17 cycle. The 1-edges form the cycle 0, 1, 2, 3 , . . . ,
16, 0, the 2-edgesform the cycle 0 , , , . . . , , , , , . . . , , ,
0, the 4-edges form the cycle 0, 4, 8, 12, 16,3, 7, 11, 15, 2, 6, 10, 14, 1, 5, 9, 13, 0, and the 8-edges form the cycle 0, 8, 16, 7, 15, 6, 14,5, 13, 4, 12, 3, 11, 2, 10, 1, 9, 0. Thus we have the following observation.
Observation 7. P (17) has a 2-factorization consisting of four cycles of length 17. An a - b - c triangle is a triangle in P (17) which consists of an a -edge, a b -edge, and a c -edge.For example, ∆(0 , ,
2) is a 1-1-2 triangle, while ∆(0 , ,
9) is an 8-8-1 triangle.
Lemma 8.
There exists an automorphism of P (17) which maps ∆(0 , , to any other tri-angle. In other words, the group of automorphisms of P (17) acts transitively on its triangles.Proof. The graph P (17) has 68 triangles as seen from the characteristic polynomial (coef-ficient of t is − σ b for b ∈ { , , . . . , } map ∆(0 , ,
2) to any1-1-2 triangle. Similarly σ ab for a ∈ { , , } , b ∈ { , , . . . , } maps ∆(0 , , , ) to any 2-2-4,4-4-8, or 8-8-1 triangle.Lemma 8 will be used to show that there are many isomorphic copies of the inducedsubgraphs of Section 4. P (17) In this section we consider two induced subgraphs G and G of P (17). When the entriesof the principal submatrices corresponding to the edges of G and G are nonzero, theirdeterminants are nonzero. This fact and the inertia bound are used to show that suchprincipal submatrices have either 3 positive and 4 negative eigenvalues, or 4 positive and 3negative eigenvalues.Let G be the following graph on 7 vertices and W a weight matrix for G . c ed ba hg f
56 71 324 W = a b a c d b c e d f g
00 0 e f h g h Observation 9. det( W ) = 2 abcg h Observation 10.
The graph G is an induced subgraph of Paley 17.
26 17 0 1412
213 0 1 5611151412 10 9 8 7 4316 emma 11. Let the product abcgh (cid:54) = 0 . • If abc > then n + ( W ) = 3 and n − ( W ) = 4 . • If abc < then n + ( W ) = 4 and n − ( W ) = 3 .Proof. Apply Equation (1) to G and W . Since α ( G ) = 3, we have that3 ≤ min { − n + ( W ) , − n − ( W ) } . Thus n − ( W ) ≤ n + ( W ) ≤
4. Since abcgh (cid:54) = 0, det( W ) (cid:54) = 0. Thus either n + ( W ) = 4and n − ( W ) = 3 or n + ( W ) = 3 and n − ( W ) = 4. Further, if abc >
0, then det( W ) > n − ( W ) = 4. On the other hand, if abc < W ) < n − ( W ) = 3.Note that Lemma 11 establishes the relationship between the sign of a triangle in P (17)and whether a principal submatrix has 4 positive or 4 negative eigenvalues. In Section 5 wewill use Lemma 8 to show that every triangle is part of an isomorphic copy of G . We thendeduce that in an optimal weight matrix for P (17) all triangles have the same sign.Let G be the following graph on 7 vertices and W a weight matrix for G . de gf hi ab c W = a a b c d b e f c e g d g h f i h i Observation 12. det( W ) = − a ef ghi Observation 13.
The graph G is an induced subgraph of Paley 17.
63 112 13 0
213 0 1 5611151412 10 9 8 7 4316
Lemma 14.
Let the product aef ghi (cid:54) = 0 . • If ef ghi < , then n + ( W ) = 3 and n − ( W ) = 4 . • If ef ghi > , then n + ( W ) = 4 and n − ( W ) = 3 .Proof. Apply Equation (1) to G and W . Since α ( G ) = 3, we have that3 ≤ min { − n + ( W ) , − n − ( W ) } . Thus n − ( W ) ≤ n + ( W ) ≤
4. Since aef ghi (cid:54) = 0, det( W ) (cid:54) = 0. Thus either n + ( W ) = 4and n − ( W ) = 3 or n + ( W ) = 3 and n − ( W ) = 4. Further, if ef ghi <
0, then det( W ) > n − ( W ) = 4. On the other hand, if ef ghi >
0, then det( W ) < n − ( W ) = 3. We now continue with the strategy stated in Section 2 by considering optimal weight matricesof P (17). Recall, a weight matrix W of G is optimal if α ( G ) = min { n − n + ( W ) , n − n − ( W ) } .The following observation can be found in [2] and follows by using the Cvetkovi´c bound on G − e . Observation 15. (Lemma 7.2 in [2]) Let G be α -critical and W an optimal weight matrixof G . Then w ij (cid:54) = 0 for all ij ∈ E ( G ) . In light of Observations 15 and 5 we have the following observation.
Observation 16.
Any optimal weight matrix W of P (17) has w ij (cid:54) = 0 if ij is an edge of P (17) . G and a weight matrix W , the sign of a triangle is the sign of the productof the entries of W corresponding to edges of the triangle of G . Lemma 17.
Let W be an optimal weight matrix for P (17) . Every triangle of P (17) has thesame sign.Proof. Let W be an optimal weight matrix for P (17). Suppose by way of contradictionthat there exists a triangle ∆ which has a different sign than ∆(0 , , W if necessary, suppose that the sign of ∆(0 , ,
2) is negative. ByLemma 8, there exists an automorphism σ of P (17) which maps ∆(0 , ,
2) to ∆ . Thus σ maps G to H , an isomorphic copy of G containing ∆ . By Observation 16 the entries of W corresponding to edges are nonzero. Thus Lemma 11 and the fact that the signs of thetriangles are different imply that the principal submatrix of W corresponding to G has 4positive eigenvalues while the principal submatrix of W corresponding to H has 4 negativeeigenvalues. By Lemma 3 and Observation 4, W is not optimal. Contradiction. Thereforeall triangles have the same sign. Lemma 18.
Let A be a symmetric n × n matrix with nonzero sub and super diagonal.There exists a (1 , − -diagonal matrix D such that the sub and super diagonals of DAD arepositive.Proof.
We proceed by induction on n . If n = 2 and a = a > a <
0. Let D be a diagonal matrix with exactly one 1 and one -1. Then the off-diagonalentries of DAD are positive.Now assume that the hypothesis is true for n = k . Let A be a symmetric matrix of order k + 1 with nonzero sub and super diagonals. Since A ( k + 1) ( A with k + 1th row and columndeleted) is a k × k symmetric matrix, there exists a (1 , − D such thatthe sub and super diagonal entries of D A ( k + 1) D are positive. If a k,k +1 = a k +1 ,k >
0, then D = D ⊕ [1]. On the other hand, if a k,k +1 = a k +1 ,k < D = D ⊕ [ − DAD will be positive.Note that in Lemma 18, D is the identity matrix. Thus DAD and A are similar andhave the same eigenvalues. In light of this fact, we may assume that any weight matrix for P (17) has positive entries on its super and sub diagonal. These entries correspond to all the1-edges of P (17) except for edge (16 , ,
0) which is negative.
Lemma 19.
Let W be an optimal weight matrix of P (17) and assume that all the 1-edgesare positive. Then all edges of P (17) are positive and the sign of every triangle is positive. roof. Let W be an optimal weight matrix of P (17). By Lemma 17, all the triangles of P (17) have the same sign. Suppose that the sign of every triangle is negative. Since every2-edge belongs to a 1-1-2 triangle and all 1-edges are positive, all the 2-edges are negative.Since every 4-edge belongs to a 2-2-4 triangle and the 2-edges are negative, all the 4-edgesare negative. Since every 8-edge belongs to a 4-4-8 triangle and the 4-edges are negative,the 8-edges are negative. But ∆(0 , ,
9) is an 8-8-1 triangle which has positive sign. Thus itis not possible for the 1-edges to be positive and have every triangle be negative.Thus the sign of every triangle is positive. Following a similar argument as above, 1-edges force 2-edges to be positive which force 4-edges to be positive which force 8-edges tobe positive. Thus all the edges of P (17) are positive as desired. Lemma 20.
Let W be an optimal weight matrix of P (17) and assume that all the 1-edgesare positive with the exception of edge (16 , . Then the remaining edges are negative with theexception of 2-edges (1 , , (0 , , 4-edges (0 , , (1 , , (2 , , (3 , , and 8-edges (0 , , (1 , , (2 , , (3 , , (4 , , (5 , , (6 , , (7 , . Further, the sign of every triangle isnegative.Proof. Let W be an optimal weight matrix of P (17). By Lemma 17, all the triangles of P (17)have the same sign. Suppose that the sign of every triangle is positive. Since every 2-edgebelongs to a 1-1-2 triangle and all 1-edges except (0 ,
16) are positive, all 2-edges are positivewith the exception of edge (1 ,
16) and (0 , ,
16) and (0 , , , (1 , , (2 , , (3 , , , (1 , , (2 , , (3 , , , (1 , , (2 , , (3 , , (4 , , (5 , , (6 , , (7 , , , ,
9) is the only negative edge of the triangle andthus ∆(0 , ,
9) has negative sign. This contradicts that the sign of every triangle is positive.Thus the sign of every triangle is negative. Note that changing the sign of the trianglesfrom positive to negative just switches the sign of all the 2-edges, 4-edges, and 8-edges in theabove argument. Therefore all the 2-edges are negative with the exception of (1 , , (0 , , , (1 , , (2 , , (3 , , , (1 , , (2 , , (3 , , (4 , , (5 , , (6 , , Theorem 21.
The Paley graph on 17 vertices has no optimal weight matrices.Proof.
Suppose by way of contradiction that W is an optimal weight matrix of Paley 17.By Lemma 18, there exists a diagonal matrix D such that the sub and super diagonalentries of DW D are positive. As D is a diagonal matrix the zero-nonzero structure of W is maintained. Further, as D is the identity, DW D is similar to W . Thus DW D is anoptimal weight matrix for P (17). Let DW D = (cid:102) W . Notice that the sub and super diagonalentries of (cid:102) W correspond to the 1-edges of P (17) except (0 , (cid:102) W corresponding to edgesare nonzero. 9irst we consider the case where (0 ,
16) is positive. By Lemma 19, all the edges of P (17)are positive. In other words all the entries of (cid:102) W corresponding to edges are positive. ByObservation 13, G is an induced subgraph of P (17). By Lemma 14, (cid:102) W has a principalsubmatrix corresponding to G which has four positive eigenvalues. On the other hand,Lemmas 19 and 11 imply that (cid:102) W has a principal submatrix corresponding to G which hasfour negative eigenvalues.Now the case where (0 ,
16) is negative and the rest of the 1-edges are positive. By Lemma20, the remaining edges are negative with the exception of 2-edges (1 , , , , , , , , , , , , , , G with vertices0, 1, 2, 3, 6, 12, and 13. Edges (0 , , , ,
3) are positive, while edge (1 ,
3) isnegative. Thus by Lemma 14, the principal submatrix of (cid:102) W corresponding to G has fournegative eigenvalues. On the other hand, Lemmas 20 and 11 imply that (cid:102) W has a principalsubmatrix corresponding to G which has four positive eigenvalues.Thus in either case (cid:102) W has principal submatrices which have four positive and four nega-tive eigenvalues. Recall (Observation 4) that α ( P (17)) = 3. By Lemma 3, (cid:102) W is not optimal.Contradiction. Thus there does not exist an optimal weight matrix of Paley 17. After having shown that the inertia bound for Paley 17 is not tight, it is natural to considerits induced subgraphs. In fact deleting a vertex from Paley 17 yields a graph which is alsonot inertia tight.
Theorem 22.
The inertia bound for the graph obtained from Paley 17 by deleting a vertexis not tight.Proof.
The argument is very similar to that of Paley 17. For the sake of brevity, we includeonly an outline. As Paley 17 is vertex transitive, the graph obtained by deleting the zerovertex is isomorphic to the graph obtained by deleting any other single vertex. Let G bePaley 17 less the zero vertex. It can be shown that α ( G ) = 3 and that G is α -critical. Thusby Observation 15 all entries of an optimal weight matrix of G corresponding to edges arenonzero.There are two key properties of the graph G in Section 4. First, if the entries of a weightmatrix corresponding to edges are nonzero, then the determinant is nonzero. Second, thesign of the determinant (and hence whether the inertia is (4 , ,
0) or (3 , , G in Paley 17, only 40 arestill present in G . However, there are various other induced subgraphs of G on 7 verticeswhich also possess the above two properties. All but 8 of the 56 triangles of G belong tosuch an induced subgraph. (Every vertex of these 8 triangles is not adjacent to 0 in Paley17.) Setting aside these 8 triangles and using the same argument found in Lemma 17, theother 48 triangles have the same sign in any optimal weight matrix for G .10uppose by way of contradiction that there exists an optimal weight matrix M of G .Using Lemma 18, we may assume that all the entries of M corresponding to the 1-edgesof G are positive. The two 1-1-2 triangles which are not part of the 48 are ∆(5 , ,
7) and∆(10 , , G , except (5 , , , , ,
7) and ∆(10 , , G , except (5 , , , , , , , , ,
11) and ∆(6 , , ,
15) and (2 , , ,
14) and ∆(3 , , G solely based on whether the 48 triangles have positive sign or negativesign.In the case that the sign of the 48 triangles are positive, all the entries of M correspondingto edges are determined to be positive. Note that a copy of graph G from Section 4 remainsin G . In the case that the sign of the 48 triangles are negative all entries of M correspondingto edges are negative except for all the 1-edges, and the edges (16 , , , , , , , , , , , G on vertices1 , , , , , ,
14, the corresponding principal submatrix will have positive determinant sincethe 5-cycle on vertices 1 , , , ,
14 has negative sign. Thus in either case M has a principalsubmatrix with 4 positive eigenvalues and another with 4 negative eigenvalues. Thereforeby Lemma 3, M is not optimal, a contradiction. The goal of this paper was to demonstrate that the answer to Question 1 is no. Having donethat, we turn our attention to some related unanswered questions.In [5] a method for weighting Cayley graphs is introduced. This method weights thegenerators of the Cayley graph which corresponds to weighting the 1-regular and 2-regularsubgraphs in its decomposition. As Paley 17 has a 2-factorization, this method weights eachof the 4 cycles of length 17. Using this method it is possible to construct a weight matrixfor Paley 17 with inertia (13 , , −
22, the 4-edges a weight of −
12, and the 8-edges a weight of 7. Sage determined thatthis matrix has 13 positive eigenvalues and 4 negative eigenvalues. Thus for Paley 17 thegap between α and min { − n + ( W ) , − n − ( W ) } is one. Question 23.
How large can the gap be in Equation 1?
It is somewhat unsatisfying that Paley 13 remains undetermined. While considering this11uestion we were able to show that any non-negative weight matrix for Paley 13 is notoptimal.
Question 24.
Does there exist an optimal weight matrix for Paley 13?
In [2] it is shown that all the induced subgraphs of Paley 13 are inertia tight. Theorem22 shows that at least one of the proper induced subgraphs of Paley is not inertia tight. Itseems likely that others are as well.
Question 25.
Which of the proper induced subgraphs of P (17) are inertia tight? We conclude with the following two questions.
Question 26.
What other methods exist for determining whether a graph is inertia tight?
Question 27.
What other methods exist for constructing weight matrices with a relativelylarge number of positive or negative eigenvalues?
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