arXiv:1403.2629v1 [math.CO] 11 Mar 2014
NEW RESULTS ON EIGENVALUES AND DEGREE DEVIATION FELIX GOLDBERG Abstract. Let G be a graph. In a famous paper Collatz and Sinogowith had proposed to measure its deviation from regularity by the difference of the (adjacency) spectral radius and the average degree: (G) = ρ(G) − 2m n . We obtain here a new upper bound on (G) which seems to consistently outperform the best known upper bound to date, due to Nikiforov. The method of proof may also be of independent interest, as we use notions from numerical analysis to re-cast the estimation of (G) as a special case of the estimation of the difference between Rayleigh quotients of proximal vectors.
1. Introduction and main result Let G be a connected graph with adjacency matrix A. Then A has a Perron value ρ and a positive Perron unit vector v that satisfy Av = ρv, ||v||2 = 1. Suppose that the graph G has n vertices and m edges. Then 2m is n equal to the average vertex degree of G. The following classic result of Collatz and Sinogowitz relates it to the Perron value: Theorem 1.1. [6] Let G be a connected graph with n vertices and m edges. Then 2m (1) ρ≥ n and equality holds if and only if G is regular. Theorem 1.1 allows us to consider (G) = ρ − 2m as a measure of n the graph’s irregularity. As such it has been studied by various authors Date: February 11, 2013. 1991 Mathematics Subject Classification. 05C50,05C07,15A42,91D30. Key words and phrases. irregularity, adjacency matrix, Perron vector, Perron value, mean degree, harmonic graph, spectral radius, Rayleigh quotient numerical analysis. This research was supported by the Israel Science Foundation (grant number 862/10.) 1
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[1, 2, 5, 13, 15]. For some alternative ways of measuring the irregularity see [8]. We also call attention to [14] where (1) is placed in a wider context. The inspiration for the present paper is given by the results of Nikiforov [13] who related (G) to two other natural measures of irregularity which are based on the degree sequence of G. These are in fact the first two moments of the degree sequence: X 2m s(G) = du − n u∈V (G)
and
1 X 2m 2 var(G) = du − , n n u∈V (G)
where du is the degree of the vertex u. As observed in [13], s(G) and var(G) are related: s2 (G) ≤ var(G) ≤ s(G). n2 Another interesting property of var(G) can be obtained from the Popoviciu inequality [17]: (∆(G) − δ(G))2 . 4 Our goal is to improve on the following result of Nikiforov: var(G) ≤
Theorem 1.2. [13] Let G be a graph. Then p var(G) √ (2) ≤ (G) ≤ s(G). 2 2m Let S = ||v||1 , that is the sum of the entries of the unit Perron eigenvector. Note that by Cauchy-Schwarz, S 2 ≤ n, with equality iff the graph G is regular, and thus S −1 may also serve as a measure of irregularity. Theorem 1.3. Let G be a connected graph. Then r p n (3) (G) ≤ var(G) · − 1. S2 The proof requires a brief detour into the field of numerical analysis, taking [19, Section 2] as our benevolent guide. Let M be a real symmetric matrix and x 6= 0 a (real) vector. The Rayleigh quotient is %(x) =
xT M x . xT x
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It is well-known that the eigenvalues of M are precisely the stationary points of %(·) : Rn → R. Suppose now that Ax = λx, so that %(x) = λ. Suppose also that y is a vector lying close to x. We can expect by the continuity of %(·) that %(x) will be close to %(y). Since the function %(·) is homogenous, the useful way to measure proximity of vectors will be by the angle between x and y: | hx, yi | ∠(x, y) = arccos . ||x||2 · ||y||2 There are two ways of making this statement precise: the a priori bound (4) and the a posteriori bound (5). The latter bound uses the residual vector r(y) = Ay − %(y)y. (4)
|λ − %(y)| ≤ (λmax (M ) − λmin (M )) · sin2 (∠(x, y)).
||r(y)|| · tan(∠(x, y)). ||y|| It is not possible to tell in advance which of the bounds will turn out more useful for a particular problem. For our purposes the a posteriori works much better, so we will henceforth focus on it. Let us take M = A and x = v and y = 1n . Then we have that λ = ρ(G) and %(y) = 2m . The residual vector r(y) is: n 2m . r(y) = Ay − %(y)y = d − n Therefore ||r(y)|| p (6) = var(G). ||y|| On the other hand, the cosine of the angle between v and 1n is: | hv, 1n i | S (7) cos ∠(v, 1n ) = =√ . ||v||2 · ||1n ||2 n The claim of Theorem 1.3 now follows from (5, 6, 7). (5)
|λ − %(y)| ≤
Remark 1.4. Extensive numerical calculations indicate that the bound of (3) is stunningly close to the true value of (G) in all cases examined. However, the actual estimation of S 2 on which the bound depends is often very difficult. Therefore, we are willing to settle for a weaker p bound: (G) ≤ var(G) which would still improve upon Theorem 1.2. This fails to be true for disconnected graphs but we strongly believe that it is true for connected graphs.
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Conjecture 1.5. If G is connected, then p (G) ≤ var(G). Clearly, the conjecture is equivalent to n S 2 ≥? . 2 2. First examples - exact computation of S 2 In order to demonstrate the strength of Theorem 1.3 we would like to consider fist a number of examples in which the Perron vector v can be easily computed explicitly, and therefore a formula for S 2 can be written down. Later we will develop some ways of estimating S 2 from below in cases where explicit expressions for v are not available or are too intimidating to be effectively used. 2.1. Bicliques. Let G = Kp,q be a complete bipartite graph, with √ p ≤ q. It is not hard to compute that ρ(G) = pq and 2pq √ (G) = pq − . q+p Nikiforov’s estimate is: s p (q − p) (G) ≤ s(G) = 2pq , q+p which has the correct order of magnitude but is off by multiplicative and additive constants. Let us now compute the bound of Theorem 1.3: p (q − p) √ var(G) = pq, q+p and the Perron vector of Jp 0 A(G) = 0 Jq is easily verified to be v= Therefore S =
√ √1 ( q 2
√1 2q √1 2p
· jp · jq
.
√ + p) and r √ √ q− p n −1= √ √ . S2 q+ p
Finally, a simple algebraic manipulation will show that in this case equality obtains in (3) and thus our bound is sharp.
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2.2. Harmonic graphs. A Pgraph is called harmonic [7, 10] if for some real λ the equality λdvi = j∼i dvj holds for all i = 1, 2, . . . , n. This is clearly equivalent to ρ = λ,
v = c · d, c > 0.
In this case we can evaluate the term S precisely. Let us define the quantity X ZG = d2v v∈V (G)
(cf. e.g. [16, 11]). Then (8)
v=
p ZG · d
for a harmonic graph. Theorem 2.1. Let G be a harmonic graph on n vertices and with m edges. Then r p nZG (9) (G) ≤ var(G) · − 1. 4m2 Proof. By (8) we have S2 =
4m2 . ZG
Example 2.2. Consider a family of 3-harmonic graphs, constructed in [3]. See Figure 1. The graph Tk has n = 3k vertices and m = 4k edges. It has k vertices of degree 4 and 2k vertices of degree 2, therefore 2m = 38 and (G) = 13 . n Nikiforov’s estimate (2) gives: r p 8k (G) ≤ s(G) = , 3 failing to flesh out the fact that (G) is constant for the whole family. On the other hand, ZG = 24k and var(G) = 89 and thus from (9): r r r p nZG 1 8 1 (G) ≤ var(G) · −1= · = . 2 4m 8 9 3 So once again, equality holds.
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Figure 1. T4 and T5 3. Estimating S 2 by the Wilf method Recall the classic result due to Wilf: Theorem 3.1. [18] Let G be a graph with clique number ω and spectral radius ρ. Then w (10) S2 ≥ ρ. w−1 Wilf’s result is in many cases sufficiently powerful to obtain, in conjunction with Theorem 1.3, excellent estimates on (G). In particular we can use it to prove our Conjecture 1.5 in a special case: Corollary 3.2. If G is a connected graph on n vertices with ω(G) ≥ n2 , then p (G) ≤ var(G). Proof. Since the spectral radius is monotone with respect to subgraphs (cf. [4, p. 33]) we have ρ ≥ ω − 1. Therefore S 2 ≥ ω ≥ n2 . In the remainder of this section we will study a particular example. 3.1. Pineapples. The pineapple graph P (n, q) consists of a clique on q vertices and a stable set on n−q vertices, so that one particular vertex in the clique in adjacent to all the vertices in the stable set. Pineapple graphs have high values of (G) and in fact have been conjectured to be its maximizers: Conjecture 3.3. [1] Among all graphs on n ≥ 10 vertices the graph G with the highest value of (G) is G = P A(n, q), q = b n2 c + 1.
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Figure 2. The graph P (10, 6) Example 3.4. Consider the graph G = P (2k, k + 1). Nikiforov’s estimate is: r p k 3 − 3k + 2 (G) ≤ s(G) = . k Since ω = k + 1 ≥ k =
n 2
we can use Corollary 3.2 to obtain:
p k−1 √ 2 var(G) = · k + 4k − 4. 2k Thus an improvement by a factor of two is gained. (G) ≤
4. Estimating S 2 for cones Let us now consider the case when ∆(G) = n − 1, i.e. when some vertex is adjacent to all other vertices. Such a vertex is called dominating or universal. Denote by H the subgraph obtained by deleting v and all edges incident upon it from G. Another common mode of speaking is to say that G is the cone over H and the notation G = H ∨ K1 is used accordingly. The pineapple graph is in fact a cone over the disjoint union of a clique and a stable set. As we have seen, for the pineapple graph the Wilf method works very well. However, in other cases, it may yield poor results. Therefore we shall now develop an alternative method of estimating S 2 specifically for cones and then illustrate its power by an example.
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Theorem 4.1. [9] Let G be a connected graph on vertices {1, 2, . . . , n} with Perron vector v, normalized so that ||v||2 = 1. For every vertex i ∈ V (G) let Hi = G − {i} be the subgraph obtained by deleting i from G and let ρHi be its spectral radius. Then for any 1 ≤ i ≤ n: 1 (11) vi2 ≥ . 1 + (ρ−ρdiH )2 i
Lemma 4.2. Let G be a graph with vertex set {1, 2, . . . , n} and suppose that 1 is a dominating vertex. Then (12)
S = (ρ + 1)v1 .
Proof. Consider the first entries of both sides of the equation Av = ρv: n X ρv1 = (Av)1 = vj = S − v1 j=2
Now, combining (11) and (12) we immediately obtain: Theorem 4.3. Let G = H ∨ K1 and let ρH be the spectral radius of H. If G has n vertices, then: (13)
S2 ≥
(ρ + 1)2 (ρ − ρH )2 . (ρ − ρH )2 + n − 1
Note that the right-hand side of (13) is a decreasing function of ρ − ρH and therefore we can estimate it from below, in turn, by using bounds of the form ρ ≥ a and ρH ≤ b, to obtain: S2 ≥
(a + 1)2 (a − b)2 . (a − b)2 + n − 1
Example 4.4. Let G = P20 ∨ K1 be the cone over the path on 20 vertices. To fairly compare the bounds on Sq2 provided by (10) and (13) Pn 2 1 we will use Hofmeister’s [12] bound ρ ≥ i=1 di for both. Since n the degrees of G are: n−1, 3 repeated n−3 times, and 2 repeated twice, we have: r 1 2 ρ≥ (n + 7n − 18) = 5.21. n Thus, (10) yields: 3 S 2 ≥ ρ ≥ 7.815, 2 whereas (13) yields, using ρH ≤ ∆(H) = 2: S 2 ≥ 13.115.
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The actual value of S 2 in this example is 16.8305 while the right-hand side of (13) is 16.5815. 5. Acknowledgments I wish to thank Dr. Clive Elphick for illuminating correspondences on the subject of this paper and Professor Martin Golumbic for a careful reading of a first draft. References [1] M. Aouchiche, F. K. Bell, D. Cvetkovi´c, P. Hansen, P. Rowlinson, S. K. Simi´c, and D. Stevanovi´c. Variable neighborhood search for extremal graphs. XVI. Some conjectures related to the largest eigenvalue of a graph. European J. Oper. Res., 191:661–676, 2008. [2] F. K. Bell. Eigenvalues and degree deviation in graphs. Linear Algebra Appl., 161:45–54, 1992. [3] B. Borovi´canin, S. Gr¨ unewald, I. Gutman, and M. Petrovi´c. Harmonic graphs with small number of cycles. Discrete Math., 265(1–3):31–44, 2003. [4] A. E. Brouwer and W. H. Haemers. Spectra of Graphs, volume 223 of Universitext. Springer, 2012. [5] S. M. Cioab˘ a and D. A. Gregory. Large matchings from eigenvalues. Linear Algebra Appl., 422(1):308–317, 2007. [6] L. Collatz and U. Sinogowitz. Spekter endlicher Grafen. Abh. Math. Sem. Univ. Hamburg, 21:63–77, 1957. [7] A. Dress and I. Gutman. The number of walks in a graph. Appl. Math. Lett., 16:797–801, 2003. [8] C. Elphick and P. Wocjan. New measures of graph iregularity. http://arxiv. org/abs/1305.3570v4, 2013. [9] F. Goldberg. A lower bound on the entries of the principal eigenvector of a graph. http://arxiv.org/abs/1403.1479, 2014. [10] S. Gr¨ unewald. Harmonic trees. Appl. Math. Lett., 15(8):1001–1004, 2002. [11] I. Gutman and K. C. Das. The first Zagreb index 30 years after. MATCH Commun. Math. Comput. Chem., 50:83–92, 2004. [12] M. Hofmeister. Spectral radius and degree sequence. Math. Nachr., 139:37–44, 1988. [13] V. Nikiforov. Eigenvalues and degree deviation in graphs. Linear Algebra Appl., 414(1):347–360, 2006. [14] V. Nikiforov. Walks and the spectral radius of graphs. Linear Algebra Appl., 418(1):257–268, 2006. [15] V. Nikiforov. Bounds on graph eigenvalues II. Linear Algebra Appl., 427(2– 3):183–189, 2007. [16] V. Nikiforov. The sum of the squares of degrees: Sharp asymptotics. Discrete Math., 307(24):3187–3193, 2007. [17] T. Popoviciu. Sur les ´equations alg´ebriques ayant toutes leurs racines r´eelles. Mathematica, Cluj, 9:129–145, 1935. [18] H. S. Wilf. Spectral bounds for the clique and independence numbers of graphs. J. Comb. Theory, Ser. B, 40:113–117, 1986.
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[19] P. Zhu, M. E. Argentati, and A. V. Knyazev. Bounds for the Rayleigh quotient and the spectrum of self-adjoint operators. SIAM J. Matrix Anal. Appl., 34(1):244–256, 2013. Caesarea-Rothschild Institute, University of Haifa, Haifa, Israel E-mail address:
[email protected] 
