The conjecture of Graham and Lovász that the (normalized) coefficients of the distance characteristic polynomial of a tree are unimodal is proved; it is also shown that the (normalized) coefficients are log-concave. Upper and lower bounds on the location of the peak are established.This article is published as Aalipour, Ghodratollah, Aida Abiad, Zhanar Berikkyzy, Leslie Hogben, Franklin Kenter, Jephian C-H. Lin, and Michael Tait. "Proof of a Conjecture of Graham and Lovasz concerning Unimodality of Coefficients of the Distance Characteristic Polynomial of a Tree." The Electronic Journal of Linear Algebra 34 (2018): 373-380. DOI: 10.13001/1081-3810.3493.</p

Aalipour, Ghodratollah

Abiad, Aida

Berikkyzy, Zhanar

Hogben, Leslie

Kenter, Franklin

Lin, Jephian

Tait, Michael

Digital Repository @ Iowa State University (ISU)

Elcctronic .Journal of Linear Algebra, ISSN 1081-3810 A publication or the lnLcrnationa.1 Linear Al_c;chra Society Volume ::\4, pp. 373-380, Auguflt 20JR. PROOF OF A CONJECTURE OF GRAHAM AND LOV Asz CONCERNINGUNIMODALITY OF COEFFICIENTS OF THE DISTANCE CHARACTERISTIC POLYNOMIAL OF A TREE* GIIODRJ\TOLLAII AALIPOlJRt, AIDA AOIADi, ZIIANAR BERIKKYZY§, LESLIE IIOGBEN1, FRANKLIN H.J. KENTERII, .IEPHIAN C.-H. LIN§, AND MICHAEL TAIT# Abstract. The conjecture of Graham aud Lov{isz that the (normalized) coefficients of the distance characteristic poly nomial of a tree arc unimodal is proved; it is also shown that the (uormali�cd) coefficients arc log-concave. Upper aud lower bouuds on the location of the peak arc established. Key words. Distance matrix, Characteristic polyuomial, Unimodal, Log-concave. AMS subject classifications. 05C50, 05C12, 05C31, 15Al8. 1. Introduction. The distance matri:1; V(G) of a simple, finite, undirected, connected graph G is thematrix indexed by the vertices of G with ( i, j)-entry equal to the distance between the vertices Vi and Vj, i.e., the length of a shortest path between Vi and Vj- The characteristic polynomial of 'D(G) is defined by Pv(a)(x) = det(:i;J - V(G)) and is ca.lied the distance characteristic polynomial of G. Since V(G) is a real symmetric matrix, all of the roots of the distance characteristic polynomial a.re real. Distance matrices were introduced in the study of a data communication problem in [9]. Thi'i problem involves finding appropriate addresses so that a message can move efficiently through a series of loops from its origin to its destination, choosing the best route at each switching point. Recently, there has been renewed interest in the loop switching problem [6). There has also been extensive work on distance spectra; sec [l] for a recent survey. A sequence a0 , a1, a2, ... , an of real numbers is 11,nimodal if there is a k such that ai-l :S ai for i :S k anda; � a;+1 for i � k, and the sequence is log-concave if a; � aj-laj+l for all j = 1, .. . , n - 1. Recent surveys about unirnodality and related topics can be found in [2, 3], and a classical presentation is given in [5]. For a gTaph G on n vertices, the coefficient in dct(V(G) - :J:J) = (-l) nPD(a)(:i:) of :r;k is denoted by Ok ( G) by Graham and Lov11sz [8), so the coefficient of xk in PD(G) (:r;) is ( -1 )no!.:( G). The following statement appears on page 83 in [8] (a tree is a connected graph that docs not have cycles, and n is its order, 1.c., number of vertices): It appears that in fact for each tree T, the q11,antities (-l)'1- 1 ok(T)/211-k-2 are 11,nimodalwith the ma.r,im11,m vafoe occ11,rrin_q fork= l¥J. We see no way to prove this, however. •R,:,-ceived by the editors on February 9, 2017. Accept,xl for publication on :June 12, 2018. Handling EdiLor: Bryan L. Shader. Corresponding Author: Leslie Hogbeu. t Department of Mathematics and Computer Sciences, Kharazmi University, 50 Taleghani St., Tehran, Iran and Department of Mathematical aud Statistical Sciences, University of Colorado Denver, CO, USA (alipour.ghodratoUah@gmail.com). tocparlmcnt of Econometrics and Operations Research, Tilburg University, Tilburg, The Netherlands ( aidaabia.<l@gmail.com). §Department of MaLhcmaLics, Iowa SLatc University, Ames, IA 50011, USA (zhauar@ucr.edu). 1Department of Mathematics, Iowa State University, Ames, lA 50011, USA, and American lnstilutc of Mathematics, 600 E. Brokaw R.oad, Sau .Jose, CA 95112, USA (hogbcu@aimaLh.org). IIMathematics Department, United State Naval Academy, Chauvcnet Hall, 572C Holloway Road, Annapolis, MD 21102, USA (frau.kli11.kc11ter@gmail.com). #Ocparlmcnt of Mathematics, University of California Sau Diego, La Jolla CA, 92037, USA (mLait@cmu.edu). Electronic .Journal of Linear Algebra, IS!-iN 1081-3810 A puhlication of the lnuirnat.ional Linear Alp;cbra Societ.y Voh11nc 3d, pp. 373-380, AuguRt 2018. G. Aalipour, A. Abiad, Z. Berikkyzy, L. Hogben, F .H .. J. Kenter, .J.C.- H. Lin, and M. TaitFACT I.I. [8, equation (44)] For a tree T on n  vertices, 374 Throughout this discussion, the order of a graph is assumed to be at least three (any sequence ao is trivially unimodal a.nd the peak location is 0). For a graph G of order n a.nd O ::::; k ::::; n - 2, define dk (G) = lbk(G)l/2n-k-2. We call the numbers dk (G) the normalized coefficients. If T is a tree, thendk (T) = (-1)11-1bk(T)/2n-k-2 by Fact 1.1. For a tree, the normalized coefficients represent counts ofcertain subforests of the tree [8]. The conjecture in [8] can be rephrased as: For a tree T of order n, the sequence of nmmalized coefficients do(T), ... , dn-2(T) is uni­modal and the peak occurs at l � J . The conjecture regarding the location of the peak was disproved by Collirn-; [4] who showed that for both stars and paths the sequence do (T), ... , dn-2(T) is unimodal, but for paths the peak is at approximately(1 - Js) n (and at l�J for stars).1 Conjecture 9 in (4], which Collins attributes to Peter Shor, is:CONJECTURE 1.2. ( Collins-Shor) The {normalized} coefficients of the distance characteristic polynomial for any tree T with n vertices are unimodal with peak between l � J and r ( 1 - )s) nl . In [4], Conjecture 9 is stated without the floor or ceiling; l � J is dearly the intended lower bound, since(4, Theorem 1] establishes l � J as the peak location for a star. An examination of the proof of [4, Theorem 3]shows that the ceiling is needed in the upper bound (although the pa.th P71 may attain either the floor or the ceiling depending on n). This conjecture is included in [l] as Conjecture 2.6 (aga.in without "normalized" and without the floor and ceiling), followed by the comment, "No more results are known ab<mt that con_jectnre." The log-concavity of the sequences dk (T) of normalized coefficients and lbk (T)I of absolute values of coefficients arc equivalent, and we show in Theorem 2.1 below that both sequences lbo(T)I, ... , lbn-2(T)Iand do (T), ... , �-2(T) arc log-concave and unimodal. In Section 3 we establish an upper bound of f �nl for the peak location of the normalized coefficients. We also show that the coefficient � can be improved when the tree is "star-like" with many paths of length 2. Further, we give a lower bound of d�l where d is the diameter of the tree (i.e., the rnunber of edges in a longest path in the tree). Finally, in Section 4 we give an example showing unimodality need not be true for gTaphs that a.re not trees. To establish these results, we need some additional definitions and facts. The next observation 1s immediate from the definition. OBSERVATION 1.3. Let ao, a1, a2, ... , an be a .sequence of real rmrnbers, let c and s be nonzero real num­bers, and define bk = sck ak. Then ao, a1, a2, ... , a,, is log- concave if and only if bo, b1, b2, ... , b,. is lo_q­concave. Consider a real polynornial p(:i;) = <LnXn + · · · + a1x + ao. The coefficient sequence of p is the sequence ao, a1, a2, ... , <Ln - The polynomial pis real- rooted if a.ll roots of pa.re real (by convention, constant polynomials arc considered real-rooted). The next result is known (sec, for example, [2, 3, 51). It is straightforward to adapt the proof of [2, Lemma 1.1] or (5, Theorem B, p. 270], which arc stated with the additional assumption that the polynomial coefficients a.re nonnegative, to the more general case. 1 Dcspilc use of lhc lerm cocfficienl lhroughoul [1], lhe sequence di:scusscd lhcre is dk (T), nol ok (T). Electronic .Journal of Linear Algebra, IS!-iN 1081-3810 A puhlication of the lnuirnat.ional Linear Alp;cbra Societ.y Voh11nc 3d, pp. 373-380, AuguRt 2018. 3 75 Proof of the Graham-Loviisz Unimodality Conjecture LEMMA 1.4. { a) if p(:i;) = anxn + · · · + a 1 :i; + ao is a real-rooted polynomial, then:.2 (.) "; > a;+1";-1 I . - 1 1 z (")2 _ ( ." ) ( ." ) 1 or J , . . .  , n - . J ,+1 J-1 (ii) The coefficient sequence ai of p is log-concave.(b) If ao, a1, a2, . .. , an is positive and log-concave, then ao, a1, a2, . .. , an is unimodal.2. Proof of Graham and Lov�1sz' unirnodality conjecture for the distance characteristicpolynomial of a tree. THEOREM 2.1. Let T be a tree of order n.(i) The coefficient sequence of the distance characteristic polynomial Pv(T)(:i;) is log-concave.(ii) The sequence IJo(T)I, ... , IJ,,_2(T)I of absolute values of coefficients of the distance characteristicpolynomial is log-concave and unimodal.( iii) The sequence do (T), ... , d,,-2(T) of normalized coefficients of the distance characteristic 110lynomialis log-concave and unimodal.Proof. Let V(T) be the distance matrix of T. Since PD(T) (:i;) is real-rooted, the coefficient sequence (-l)nJo(T), ... , (-l)"J,,-2(T), 0, 1 is log-concave by Lemma. 1.4 (a) (i). Therefore, the sequence (-l)"'Jo(T), ... ,(-l)"'lln-2(T) is log-concave. By Fact 1.1, (-l)n-lJk(T) >0 for O :S: k :s; n - 2, so we have (-1 )''Jk(T) < 0 for O :S: k :s; n - 2. Because all of the terms (-l)nJo(T), ... , (-l)"'Jn-2(T) a.re negative, the sequence of their absolute values { IJk(T)I} z::� is log-concave and positive. Then by Lemma 1.4 (b), the sequence IJo(T)I, ... , IJn-2(T)I is unimodal. Since dk(T) = {z,,1_2) 2k 1Jk(T)I, the log-concavity of the sequence {dk(T)}Z::� then follows from Obser-vation 1.3. Since {dk(T)}Z=:� is positive, it is llllimodal by Lemma 1.4(6). D 3. Bounds on the peak location. For a tree T of order n, the question of the location of the peak ofthe unimodal sequence of normalized coefficients { dk(T)} �=� remains open. Note that Conjecture 1.2 says that the peak location is between L0.5nj a.nd roughly ro.5528nl Computations on Sage [10, 11] confirm this conjecture for all trees of order at most 20. ln this section we show that the peak location is at most ro.6667nl for all trees of order n, and at lea.st l 't+� J for a. tree of diameter d and order n. Furthermore, the upper bound we establish is better for a. "star-like" tree, that is, when the tree has a. high fraction of the number of paths of length 2 in a. star (which attains the maximum possible number of paths of length 2). OBSERVATION 3.1. Let T be a tree on n vertices and define1 Pr(x) = - 2,,_2 det( 2xl - V(T)) .Then Pr(:i;) is a real-rooted polynomial urith coefficients -4 for x", 0 for x"-1, and dk(T) > 0 for xk when0 :S: k :S: n - 2. LEMMA 3.2. Let ao, a1, a2, ... , <Ln-2 be a unimodal sequence with ai > 0 for i = 0, ... , n - 2 such thatL�=O akxk is a real-rooted polynomial.1. If for some index .i =/ n, n - 1n -J a1 ----·-<l,n(.j + 1) ao Electronic .Journal of Linear Algebra, IS!-iN 1081-3810 A puhlication of the lnuirnat.ional Linear Alp;cbra Societ.y Voh11nc 3d, pp. 373-380, AuguRt 2018. G. Aalipour, A. Abiad, Z. Bcrikkyzy, L. Hogbcn, F.H .. J. Kenter, .J.C.-H. Lin, and M. Taitthen the peak location is at most j . 2. if for some index .i =/- n, n - 1, 0then the peak location is at least .i . (n - 2)(n - .i + 1) an-2 . . ·-->1,3.7 an-3Proof. By Lemma 1.4 (a) (i), Then a2 > ('.;)2(j + l)(n -.i + 1) J _ ( ." ) ( _n ) aj+laj-1 = ,·en_ .) <Lj+laj-1• J+l J-1 .7 .7 .i(n -j) aj (j + l)(n-j + 1) · aj-1 < (-j-. .i - 1 ... �)j+l .i 2 n - .7 a1 n(j + 1) ao ( n -j n -.i + 1 n - 1) a1 n-j+l n-.i+2 n a0 If this value is smaller than 1, then aj+l < aj a.nd the peak location is at most j.Similarly, _!!L > (j + l)(n -.i + 1) . <Lj+1 <Lj-1 - j(n -.i) aj 2: (·i+ 1 . � + 2 ... n -�) (n -.i + 1 . n � .i ... �) a71_2 .7 .7 + 1 n -;3 n -.7 n -.7 - 1 3 an-3 (n - 2)(n - j + 1) aTl-2 3.i a,._3 376 If this value is greater than 1, then aj > aj-l and the peak location is at lca8t j. □ THEOREM 3.3. Suppose T is a tree on n 2: ;3 vertices with at least p(n:;-1) paths of length 2 for somenonnegative real nnmber p. Then the peak location of the normalized coefficients do(T), d1 (T), ... , <L,,-2(T) is at most r ;=:nl. Since p = 0 applies to every tree, the peak location is at most rnnl for every tree on nvertices. Proof. By Observation 3.1, we may apply Lcrmna 3.2 to P.r(x). When O:::; .i :::; n - 2 and n-j d1 (T) ------<l n(j + 1) do(T) the peak location is at mo8t .i- Since do(T) and d1 (T) arc both positive numbcr8, the inequality is equivalent to . r n-n n2 +n .1>---=n----, n+r n+r dr (T) where r = --do(T). The formula do (T) = n - 1 is given in [9, Theorem 3]. Defining N p_1 (T) to be the number of 8ubtrcc8 of T that arc isomorphic to the path P.1 on three vcrticc8 (of length 2), the formula d1 (T) = 2n(n - 1) -Electronic .Journal of Linear Algebra, IS!-iN 1081-3810 A puhlication of the lnuirnat.ional Linear Alp;cbra Societ.y Voh11nc 3d, pp. 373-380, AuguRt 2018. 377 Proof of the Graham-Loviisz Unimodality Conjecture 2Np_,(T) -4 follows from (7, Theorem 4.1]2 by using the definition dk (T) = (-I)'•-1 ok (T)/2"-k-Z_ Since ½ p ( n  -l)(n -2) = Np3 (T) 2'. n -2, Now 2n(n - I) -2Np_,(T) -4 2n(n -1) -p ( n  - I)(n - 2) - 4 r = -------- = ----------- < (2 - p)n + 2p. n-1 n-1 n2 + n n2 + n n + I n 2 - p n ---- < n --,-----,---- = n -------,------,- < n --- = - - n. n + r (3 -p)n + 2p 3 -p + (2p/n) - 3 -p 3 -p The last inequality follows from 3�P :S (Zpin), which is justified by p :S 1.Therefore, .i = r ;=:nl is an upper bound of the peak location. □REMARK 3.4. If the number NP., (T) of paths of length two is known for every tree T in a particular family, then p can be set equal to (:�<,)). For example, for the star S,. on n vertices, NP., (Sn) = ('';1), sop = I and r �nl = I� l- Thus, for a star, our upper bound is equal to (if n is even) or one more than (ifn is odd) the known value l � J for the peak of the normalized coefficients for Sn [4, Theorem I]. We will utilize a technique similar to the upper bound in order to derive a lower bound. However, we need the following lemma to provide an estimate for the necessary ratio. LEMMA 3.5. For any tree T on n vertices with diameter d Proof. Let V := V(T) denote the distance matrix ofT, and let Vij denote its ij-cntry. From (7, equations (4c) and ( 4d)], and On-2(T) = (-1)''-1 L'Dtji<j On-:i(T) = (-l)n-l L 2Vijvjkvki ·i<j<k We will now express the corresponding normalized coefficients in terms of the traces of powers of V. First, let us consider d,,-2(T). Since the diagonal entries of V arc all zero, 2Our uotatiou is slightly dillereut but exa.miuatiou of (7, Table 2) clarifies the uota.tiou. Electronic .Journal of Linear Algebra, IS!-iN 1081-3810 A puhlication of the lnuirnat.ional Linear Alp;cbra Societ.y Voh11nc 3d, pp. 373-380, AuguRt 2018. G. Aalipour, A. Abiad, Z. Bcrikkyzy, L. Hogbcn, F.H .. J. Kenter, .J.C.-H. Lin, and M. Taitwhere the second equality follows from V being symmetric. Similarly, for d:i(T), 378 where the third line follows because if any two of i, .i, k arc equal, then the corresponding entry in V is 0. Let A1 :S: A2 :S: · · · :S: A11 =: Amax denote the eigenvalues of V(T). Since tr(V2 ) = L Af and similarly tr(V:i) = LA?, we have dn-:i(T) dn-2(T)where the la.st inequality comes from that the the row sums of V a.re bounded above by nd. □THEOREM 3.6. Let T be a tree on n 2: 3 vertices with diameter d. Then, the peak location of thenormalized coefficients do(T),d1(T), ... A,-2(T) is at least l't+:�j. Proof. By Observation 3.1, we may apply Lemma 3.2 to fr(x). When 1 :S: .i :S: n - 2 and (n - 2)(n - .i + 1) . d.,-2(T) > 1 3.i d11_;i(T) 'the peak location is at least j. Since dn-2(T) and d.,-:i(T) arc both positive numbers, the inequality is equivalent to (n - 2)(n + 1) j < (n - 2) + (3/r)' d,,_2(T)where r = ---. d,,_;i(T)By applying Lemma 3.5, � < nd. Thus, S . l"-2J o, .7 = 1+d (n - 2)(n + 1) (n - 2)(n + 1) �-��-- > �----�(n - 2) + (3/r) (1 + d)n - 2 n-2 n+l ---· 1 + d n - 2/(1 + d) n-2> l+d·is a lower bound of the peak location. □Electronic .Journal of Linear Algebra, IS!-iN 1081-3810 A puhlication of the lnuirnat.ional Linear Alp;cbra Societ.y Voh11nc 3d, pp. 373-380, AuguRt 2018. 379 Proof of the Graham-Lov1isz Unimodality Conjecture 4. Graphs that are not trees. Since the diHtance matrix of any gTaph G iH a real Hyrrnnetric matrix,the coefficient Hequence of the diHtance cha.ractcriHtic polynomial of G iH log-concave. However, it need not be the caHc that all cocfficicntH of the diHtancc charactcriHtic polynomial have the H11mc Hign. ThuH, Hta.tcmcntH analogous to thoHc in Theorem 2.1 can be fa.lHc for graphs that arc not trees. EXAMPLE 4.1. The normalized coefficients and absolute valueH of the cocfficicntH of the diHtancc charac­tcriHtic polynomial a.re not unimodal (and hence not log-concave) for the Hcawood graph H Hhown in Fig11rc 1. The coefficients of the distance characteristic polynomial arc log-concave but not unimodal..FIGURE 1. The Hcawood graph 11. The diHtancc characteristic polynomial of H iH Pv(I-I)(x) = :r:14 - 441:r:12 - 6328x11 - 36456:r:10 - 75936:r:9 + 104720x8 + 573696x7 - 118272.7;6 - 1885184:1:5 + 973056x4 + 2795520:r:3 - 3885056:r:2 + 1892:352x - 331776.The valucH of dk(H), fork= 0, ... , 12, arc 81, 924, 3794, 5460, 3801, 14728, 1848, 17928, 6545, 9492, 9114, 3164, 441. Acknowledgment. We thank Ben Braun, Steve Butler, Jay Cummings, JcHsica De Silva, Wei Gao, and Kristin HcysHc for stimulating diHcuHsionH, and gTatcfully acknowledge financial Hupport for this rcHca.rch from NSF 1500662, Elsevier, and the International Linear Algebra Society. We thank the Graduate ReHea.rch Workshop in Combinatorics (GRWC) where this reHearch took place a.nd the lnHtitute for MathernaticH and its ApplicationH (IMA) where conncctionH essential to this rcHcarch were built. RE.FERENC ES [1) M. Aouchichc and P. Hansen. Distance spc.-ctra of g;raphs: A survey. Linear Algebra Appl., 158:301-386, 2011. [2) P. Briiudcu. Orumodality, Jog-coucavity, real-rootedness, and beyond. Iu: M. Bona (editor), llandbook of t:num erative Combinatorics, CRC Pres , Boca Ratou, 2015. [3) B. Braun. Ouimodality problem:; iu Ehrhart theory. lu: A. Beveridge, .I. Griggs, L. IIogbeu, G. Musiker, and P. Tetali (c.'<iitors), Recent 'l'rnncL� in Combinator-ic.s, IMA Vol. Math. Appl., Springer, Switzerland, 159:687-711, 2016. (1) K.L. Collins. Ou a conjecture of Graham and Lov{isz about distance matrices. Discrete Appl. Math., 25:27-35, 1()89.[5) L. Comtet. Advanced Combinatorics: The Art of Finite arid infinite t:xpansions. Reidel P ublisbing Co., Dordrccht, 197'1.[6) P. Diacouis. i<'rom loop switching to graph embedding. Pleua.ry talk at Conuc.-ctious in Discrete Mathematics: i\. celebration of the work of Ron Graham. Simon .Fraser University Vancouver, BC, Canada (abstract available at http://sitcs. google.com/site/connectionsindiscrctemath), June 15-19, 2015. [7) M. Edclberg, M.R. Garey, and R.L. Graham. Ou the distance matrix of a tree. Discrete Math., 11:23-::19, 1976. [8) R.L. Graham and L. Lov,isz. Distance matrix polynomials of trec.-s. Adv. Math., 29:60-88, 1978. (9) R.L. Graham and II.O. Pollak. Ou the addn,&;iug problem for loop switching. Bell Syst. 1ix.h. J., 50:2195--251[), 1971. Electronic .Journal of Linear Algebra, IS!-iN 1081-3810 A puhlication of the lnuirnat.ional Linear Alp;cbra Societ.y Voh11nc 3d, pp. 373-380, AuguRt 2018. G. Aalipour, A. Abiad, Z. Berikkyzy, L. Hogben, F.H .. J. Kenter, .J.C.-H. Lin, and M. Tait 380 [10) J .C.-H. Lin. Sage co<lc for determining the peak of the uuimo<lal normalized coefficients of U.1c dista11cc characteristic polynomial of a lrce. Sage worksheet published 011 lite Iowa Stale Sage server al hllps:/ /sagc.malh.i,,slale.edu/home/ pub/11/ (PDF available at hllp://orion.malh.iaslaLc.cdu/lhogbcn/TrecPcak.pdf). [11) W. Stein. Sage: Open Source Malhcmalicat Soflwar-e. The Sage Group, hllp://www.sagemalh.org. 

Proof of a Conjecture of Graham and Lovász concerning Unimodality of Coefficients of the Distance Characteristic Polynomial of a Tree

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Proof of a Conjecture of Graham and Lovász concerning Unimodality of Coefficients of the Distance Characteristic Polynomial of a Tree

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