AKCE INTERNATIONAL JOURNAL OF GRAPHS AND COMBINATORICS

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1 Reprinte from AKCE INTERNATIONAL JOURNAL OF GRAPHS AND COMBINATORICS AKCE Int. J. Graphs Comb., 10, No. 3 (2013), pp In memoriam: Joan Gimbert ( ) Josep Cone, Nacho López an Josep M. Miret Departament e Matemàtica Universitat e Lleia Catalonia, Spain {jcone,nlopez,miret}@matematica.ul.cat

2 AKCE Int. J. Graphs Comb., 10, No. 3 (2013), pp In memoriam: Joan Gimbert ( ) Josep Cone, Nacho López an Josep M. Miret Departament e Matemàtica Universitat e Lleia Catalonia, Spain {jcone,nlopez,miret}@matematica.ul.cat Abstract On the first of March 2012, Joan Gimbert, an excellent colleague an mathematician, passe away at the age of 49 years. His research is evote to Graph Theory making important contributions mainly focuse on the egree/iameter problem an other istance relate problems on graphs. Keywors: Almost Moore igraphs, eccentric igraphs, bipartite an multipartite igraphs, raial Moore graphs Mathematics Subject Classification: 05C20, 05C35, 05C50, 05C12. Joan Gimbert was born in Binefar, near Lleia, Spain. He receive the egree in Mathematics from the Universitat e Barcelona (UB) in 1985 an his PhD egree in Mathematics from the Universitat Politècnica e Catalunya (UPC) in 2000, with the PhD thesis supervise by professor Miquel Àngel Fiol an entitle Applications of the spectral theory to the stuy of ense igraphs. Partially supporte by Spanish Project TIN an by Catalan Grant 2009SGR-442

3 338 In memoriam: Joan Gimbert ( ) Joan Gimbert was incorporate into teaching work in His eication an concern were constant from the first ay. Joan Gimbert joine the Polytechnic School of the Universitat e Lleia (UL) in the course as a teacher of Computer Science Enginyeering, when the egree ha only been running a year. He prepare teaching materials with great effort an perseverance. Joan Gimbert was very active an ha plenty of initiatives an projects. Little by little, several research lines were set up with which we felt at ease an from which the Lleia research group on Cryptography an Graphs emerge. The research area which arouse most interest an enthusiasm in Joan Gimbert was Graph Theory, which allowe him to establish many close frienships, collaborating with researchers aroun the worl an making important contributions mainly focuse on the egree/iameter problem an other istance relate problems on graphs. 2. Joan Gimbert s contributions to Graph Theory Joan Gimbert s first steps into Graph Theory were in the stuy of the egree/iameter problem. This is an optimization problem motivate by the moelization of interconnection networks by means of graphs. In this context, given the values of the maximum egree an the iameter of a graph (igraph), there is a natural boun for the number of vertices of it, calle Moore boun. Graphs attaining such a boun are referre to as Moore graphs. The fact that there are very few of them suggeste the relaxation of some of the constraints involving the three parameters. This le to the stuy of graphs an igraphs with small efect; that is, whose orer is close to the unattainable Moore boun. Figure 1: Joan lecturing on igraphs

4 J. Cone, N. López an J. M. Miret 339 The classification of such extremal graphs (igraphs), for certain values of the efect an iameter, by using algebraic an spectral techniques, is one of Joan Gimbert s contributions. Further Gimbert s contributions were relate to istance in igraphs. More precisely, he focuse on the istance from a particular vertex to the farthest vertex of that one, that is, the eccentricity, characterizing the eccentricity sequence of a igraph. In the rest of the section we present Joan Gimbert s main contributions to Graph Theory. Thanks to the great number of collaborators it was ifficult to inclue all that material. Thus, although we ha no intention of giving an exhaustive review of all his publications, our aim was to consier as many of them as possible Almost Moore igraphs Digraphs having maximum out-egree, iameter k an whose orer is one less than the maximum possible (efect 1), that is, N = M D (, k) 1, where M D (, k) = k, are calle almost Moore igraphs or more precisely (, k) -igraphs. In this context, Miller, Gimbert, Širáň an Slamin prove in [25] that any extremal igraph turns out to be regular. An Joan Gimbert, in his first publication [14], introuce spectral techniques to stuy the existence an classification of these igraphs. A (, k) -igraph G has the property that for each vertex there exists only one vertex, enote by r(v) an calle repeat of v (see [1]), such that there are exactly two v r(v) walks of length less than or equal to k (one of them must be of length k ). If r(v) = v, which means that v is containe in exactly one k -cycle, v is calle selfrepeat of v (see Figure 2). 0 1 v... v... v = r(v) k 1. r(v) k... r(v)... r(v) repeat of v v selfrepeat Figure 2: Local structure of a (, k) -igraph Seeing r as a permutation of the set of vertices that for each vertex v assigns its repeat r(v), it has a cycle structure which correspons to its unique ecomposition in isjoint cycles. Such cycles are calle permutation cycles of G an the number of permutation cycles of G of each length i N is enote by m i.

5 340 In memoriam: Joan Gimbert ( ) The (0, 1) -matrix P = (p ij ) associate with the permutation r an given by p ij = 1 if r(v i ) = v j is relate with the ajacency matrix A of G by the equation where J enotes the all-one matrix. I + A + A A k = J + P, The factorization of et(xi (J + P )) in Q[x] in terms of the cyclotomic polynomials Φ n (x) is given by (see [2]) where m(n) = n i m i multiple of n. et(xi (J + P )) = (x (N + 1))(x 1) m(1) 1 N n=2 Φ n (x) m(n), represents the total number of permutation cycles of orer a The approach that Joan Gimbert followe to search for (, k) -igraphs was: Fin the factorization of the characteristic polynomial et(xi A) of an almost Moore (, k) -igraph in terms of its permutation cycle structure (m 1,..., m N ). Compute tr A,..., tr A k 1 from the previous factorization. Obtain conitions for m 1,..., m N from tr A = = tr A k 1 = 0. Construct the set F of feasible structures. If F =, then almost Moore (, k) - igraphs o not exist. In this way, Joan Gimbert in his PhD thesis complete the classification of almost Moore igraphs of iameter 2 by stuying the factorization of the characteristic polynomial et(xi A) = (x )(x + 1) a 1 x a 2 (x 2 + 1) b 1 (x 2 + 2x + 2) b 2 where F n,2 (x) = Φ n (1 + x + x 2 ), a 1 + a 2 = 1 n N m n, 2 n N, n 4 b 1 + b 2 = m(4), (F n,2 (x)) m(n) 2, an by solving the system etermine by tr A = tr A 2 = 0. The main results Joan Gimbert obtaine were publishe in [14, 15], showing that line igraph L K +1 of the complete igraph K +1 is the unique (, 2) -igraph, if the egree 3. In Figure 3 shows the line igraph L K 5. For iameters greater than 2, in orer to etermine the factorization of the characteristic polynomial of the ajacency matrix A, Joan Gimbert propose the following conjecture concerning the irreucibility of the polynomials F n,k (x) = Φ n (1 + x + + x k ) : { n (k + 2) if k is even F n,k (x) is reucible in Q[x] n 2(k + 2) if k is o an n is even

6 J. Cone, N. López an J. M. Miret 341 Figure 3: Example of a (, 2) -igraph in which cases F n,k (x) has just two factors. Later, in [8], together with Cone, González, Miret an Moreno, his conjecture has been prove for iameter 3 showing as a consequence the nonexistence of almost Moore igraphs of iameter 3. Concerning the unirecte case, the Moore boun is given by M G (, k) = ( 1) + + ( 1) k. Thus, a graph with maximum egree an iameter 2 has efect 2 if its orer is 2 1. Cone an Gimbert in [7] prove, by using algebraic an spectral techniques, that for all values of the egree within a certain range 4 < 50, these graphs with iameter 2 an efect 2 o not exist. Furthermore, they formulate an analogous conjecture to the irecte case on the irreucibility of some polynomials in Q[x] an, assuming this conjecture, they showe the nonexistence of the graphs with iameter 2 an efect 2 for any egree > 5. Mixe Moore graphs generalize both unirecte an irecte Moore graphs. In [26], Nguyen, Miller an Gimbert prove that all known mixe Moore graphs of iameter 2 are unique an that mixe Moore graphs of iameter k 3 o not exist.

7 342 In memoriam: Joan Gimbert ( ) 2.2. Raial Moore graphs Raial Moore graphs appear into the context of Moore graphs. Since the iameter of a Moore graph is equal to its raius, one can consier graphs in which the conition on the iameter is relaxe just by one, while the conition on the raius is maintaine. More precisely, raial Moore graphs are regular graphs of egree, raius k, iameter k + 1 an orer the Moore boun M G (, k). There are exactly five raial Moore graphs of egree = 3 an raius k = 2 (one of them is the unique cubic Moore graph: the Petersen graph). Capevila, Cone, Exoo, Gimbert an López in [3] propose ifferent classifications of raial Moore graphs accoring to their proximity to extremal properties of a Moore graph. They also enumerate all raial Moore graphs for the cases (, k) {(3, 2), (3, 3), (4, 2)} an ranke them. In particular, the closest raial Moore graph to a Moore graph for each one of that (, k) cases was foun (see Figure 4). (3, 2) (3, 3) (4, 2) Figure 4: Several raial Moore graphs for ifferent values of (, k). It is not ifficult to fin raial Moore graphs of raius k = 2 (iameter 3) an any egree 3. However, the problem for k 3 is much more complicate. Exoo, Gimbert, López an Gómez in [9] prove the existence of raial Moore graphs of raius three. They also settle the existence problem for (, k) equal to (3, 4), (4, 4), (5, 4), an (3, 5). It is conjecture that, except for some small values of k, raial Moore graphs o not exist, but a proof is not yet in sight. Regaring the irecte case, Knor [23] prove that these extremal igraphs o exist for any value of 1 an k 1. In [19], Gimbert an López introuce a igraph operator, base on the line igraph, which allowe to construct new raially Moore igraphs an recover the known ones Multipartite igraphs Some other aspects in which Joan Gimbert contribute eal with the stuy of bipartite igraphs. It is known [13] that orer N of a strongly connecte bipartite igraph in terms

8 J. Cone, N. López an J. M. Miret 343 of its iameter k an the maximum out-egrees ( 1, 2 ) of its partite sets of vertices is boune by the Moore-like boun M( 1, 2, k) = { 2( ( 1 2 ) m ) if k = 2m + 1, ( )( ( 1 2 ) m 1 ) if k = 2m. Figure 5 shows how to erive the Moore-like boun for even an o iameter. Figure 5: Local structure of a bipartite igraph. It has been prove that, when 1 2 > 1, the igraphs attaining such a boun only exist when 2 k 4. By using the theory of circulant matrices an the so-calle De Bruijn near factorizations of cyclic groups, Fiol, Gimbert, Gómez an Wu in [11] presente some new constructions of Moore bipartite igraphs of iameter k = 3 an composite out egrees. Moreover, it is shown that any Moore bipartite igraph of iameter 4 is the line igraph of a Moore bipartite igraph of iameter 3. In a previous work [24], Gimbert an Wu gave conitions for some particular (0, 1) -matrix equations to ensure that the (0, 1) -matrix solutions have an unerlying line igraph structure. In [10], Fiol an Gimbert consier almost Moore bipartite igraphs, that is, bipartite igraphs which miss the Moore-like boun by just one vertex in each partite set. They give some necessary conitions for the existence of such igraphs in the case of o iameter. As a consequence, it is shown that any almost Moore bipartite igraph of iameter 5 is the secon orer line igraph of a Moore bipartite igraph of iameter 3. The stuy of bipartite igraphs can be extene to multipartite igraphs, assuming that every vertex of a given partite set is ajacent to the same number of vertices in each of the other inepenent sets. Comellas, Fiol, Gimbert an Mitjana in [4, 5, 6] introuce the concept of weakly istance regular an stuie some of its main properties. In [12], Fiol, Gimbert an Miller etermine when a Moore multipartite igraph is weakly istance regular. Within this framework, some necessary conitions for the existence of a Moore r -partite igraph with interpartite out-egree δ > 1 an iameter k = 2m are obtaine. In the case δ = 1, which correspons to almost Moore igraphs,

9 344 In memoriam: Joan Gimbert ( ) a necessary conition in terms of the permutation cycle structure is erive. Also, some constructions of ense multipartite igraphs of iameter two that are vertex-transitive were presente Eccentric igraphs Joan Gimbert s contributions were also relate to eccentricities in graphs an igraphs. The eccentricity of a vertex u is the maximum istance from u to the rest of the vertices of the graph. The computation of the eccentricity of all of the vertices of a graph, that is, its eccentricity sequence, is easy. Nevertheless, the converse problem, which consists in etermining if given a sequence of positive integers there is a graph with such an eccentricity sequence, becomes ifficult. The characterization of some integer sequences as eccentricity sequences of igraphs were stuie in [18] in Nacho López s PhD thesis supervise by Joan Gimbert. They foun necessary conitions for a positive integer sequence S : s 1, s 2,..., s n to be an eccentricity sequence of a igraph an characterize the eccentricity behaviour of those sequences S such that max(s) = n 1 an min(s) n 3. As an application, they prove that the sequence S : 4, 4, 5, 6, 6, 6, 6 is the first shortest sequence, given in lexicographic orer, satisfying the necessary conitions which is not the eccentricity sequence of any igraph. a b a b a b a b c c c c G ED(G) ED 2 (G) ED 3 (G) Figure 6: The iterate sequence of eccentric igraphs for a igraph G. Joan Gimbert was intereste in another relate problem regaring eccentricities in igraphs: The eccentric igraph ED(G) of a [i]graph G has the same vertex set as G but with an arc from a vertex u to a vertex v if an only if v is a farthest vertex from u in G, that is, v is an eccentric vertex of u. Again, it is easy to compute the eccentric igraph of a [i]graph, but the converse problem is much more ifficult. Gimbert, López, Miller an Ryan in [20] gave a characterization of those [i]graphs which are an eccentric igraph of some [i]graphs. Given a positive integer k, the k th iterate eccentric igraph of G is efine as ED k (G) = ED(ED k 1 (G))

10 J. Cone, N. López an J. M. Miret 345 where ED 0 (G) = G. Another of Joan Gimbert s contributions together with Miller, Ruskey an Ryan in [22] was concerning the iterate sequence of eccentric igraphs (see Figure 6) G, ED(G), ED 2 (G),.... In particular, Gimbert, López, Miller an Ryan in [21] show that, for almost every igraph G, this sequence has perio two an tail zero, that is, G = ED 2 (G) = ED 4 (G) = an G ED(G) = ED 3 (G) =. On the other han, in [27], Villaró, Gimbert an Béjar presente an algorithm that, given a sequence an just by oing arithmetic operations, provies a stratifie graph with such an eccentricity sequence. 3. Joan, the colleague an frien Joan s lifestyle goes beyon his resarch. Respectful an frienly to everyone, he was always willing to listen an help. As a teacher, Joan was much appreciate by his stuents. He knew how to encourage them; his classes were iactic, interesting an fille with useful suggestions; transmitting the great enthusiasm he felt for mathematics an for all he explaine. Joan ha many other interests. The promotion of mathematics: he publishe several articles in informative journals about prime numbers an cryptography [16] or about Google pagerank [17]. His concern an eication to social an environmental problems in which he was involve, an his passion for nature which he mae us share with him. He was always reay to organize a weeken excursion to the mountains or a bike rie on the paths of the surrouning countrysie... His eparture, which has come too soon, leaves a huge voi in all of us. References [1] E.T. Baskoro, M. Miller an J. Plesník, On the structure of igraphs with orer close to the Moore boun, Graphs Combin., 14 (1998), [2] E.T. Baskoro, M. Miller, J. Plesník an Š. Znám, Regular igraphs of iameter 2 an maximum orer, Australa. J. Combinatorics, 9 (1994), [3] C. Capevila, J. Cone, G. Exoo, J. Gimbert an N. López, Ranking measures in raial Moore graphs, Networks, 56(4) (2010), [4] F. Comellas, M.A. Fiol, J. Gimbert an M. Mitjana, On the spectrum of weakly istance-regular igraphs, Electron. Notes Discrete Math., 10 (2001),

11 346 In memoriam: Joan Gimbert ( ) [5] F. Comellas, M.A. Fiol, J. Gimbert an M. Mitjana, Weakly istance-regular igraphs, J. Combin. Theory Ser. B, 90(2) (2004), [6] F. Comellas, M.A. Fiol, J. Gimbert an M. Mitjana, On the Weak istance-regularity of Moore-type igraphs, Linear Multilinear Algebra, 54(4) (2006), [7] J. Cone an J. Gimbert, On the existence of graphs of iameter two an efect two, Discrete Math., 309(10) (2009), [8] J. Cone, J. Gimbert, J. González, J. Miret an R. Moreno, Nonexistence of almost Moore igraphs of iameter three, Electron. J. Combin. 15 (2008) #R87. [9] G. Exoo, J. Gimbert, N. López an J. Gómez, Raial Moore graphs of raius three, Discrete Appl. Math., 160,(10-11) (2012), [10] M.A. Fiol an J. Gimbert, On almost Moore bipartite igraphs with o iameter, Congr. Numer., (2001), [11] M.A. Fiol, J. Gimbert, J. Gómez an Y. Wu, On Moore bipartite igraphs, J. Graph Theory, 43(3) (2003), [12] M.A. Fiol, J. Gimbert an M. Miller, Multipartite Moore igraphs, Linear Algebra Appl., 419(1) (2006), [13] M.A. Fiol an J.L.A Yebra, Dense bipartite igraphs, J. Graph Theory, 14 (1990), [14] J. Gimbert, On the existence of (, k) -igraphs, Discrete Math., (1 3) (1999), [15] J. Gimbert, Enumeration of almost Moore igraphs of iameter two, Discrete Math., 231 (2001), [16] J. Gimbert, Do Mathematics working with prime numbers, (Catalan) Materials Matemàtics 7 (2009), 30. [17] J. Gimbert, The mathematics of Google: the PageRank algorithm, (Catalan) Butl. Soc. Catalana Mat. 26(1) (2011), [18] J. Gimbert an N. López, Eccentricity sequences an eccentricity sets in igraphs, Ars Combin., 86 (2008), [19] J. Gimbert an N. López, On the construction of raially Moore igraphs, Ars Combin., 110 (2013), [20] J. Gimbert, N. López, M. Miller an J. Ryan, Characterization of eccentric igraphs, Discrete Math., 306(2) (2006),

12 J. Cone, N. López an J. M. Miret 347 [21] J. Gimbert, N. López, M. Miller an J. Ryan, On the perio an tail of sequences of iterate eccentric igraphs, Bull. Inst. Comb. Appl., 56 (2009), [22] J. Gimbert, M. Miller, F. Ruskey an J. Ryan, Iterations of eccentric igraphs, Bull. Inst. Comb. Appl., 45 (2005), [23] M. Knor, A Note on Raially Moore Digraphs, IEEE Trans. Comput., 45(3) (1996), [24] J. Gimbert an Y. Wu, The unerlying igraph structure of some (0, 1) -matrix equations, Discrete Appl. Math., 116(3) (2002), [25] M. Miller, J. Gimbert, J. Širáň an Slamin, Almost Moore igraphs are iregular, Discrete Math., 218 (2000), [26] M.H. Nguyen, M. Miller an J. Gimbert, On mixe Moore graphs, Discrete Math., 307(7-8), (2007), [27] E. Villaró, J. Gimbert an R. Béjar, Eccentric sequences in stratifie graphs, Bull. Inst. Comb. Appl., 59 (2010),