145 research outputs found

    On the randic index of graphs

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    © 2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/For a given graph G = (V, E), the degree mean rate of an edge uv ¿ E is a half of the quotient between the geometric and arithmetic means of its end-vertex degrees d(u) and d(v). In this note, we derive tight bounds for the Randic index of G in terms of its maximum and minimum degree mean rates over its edges. As a consequence, we prove the known conjecture that the average distance is bounded above by the Randic index for graphs with order n large enough, when the minimum degree d is greater than (approximately) ¿1/3 , where ¿ is the maximum degree. As a by-product, this proves that almost all random (Erdos–Rényi) graphs satisfy the conjecturePeer ReviewedPostprint (author's final draft

    Iterated line digraphs are asymptotically dense

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    © 2017. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/We show that the line digraph technique, when iterated, provides dense digraphs, that is, with asymptotically large order for a given diameter (or with small diameter for a given order). This is a well- known result for regular digraphs. In this note we prove that this is also true for non-regular digraphsPostprint (author's final draft

    The spectra of subKautz and cyclic Kautz digraphs

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    © 2017. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/Kautz digraphs K(d,l) are a well-known family of dense digraphs, widely studied as a good model for interconnection networks. Closely related with these, the cyclic Kautz CK(d,l) and the subKautz sK(d,2) digraphs were recently introduced by Böhmová, Huemer and the author. In this paper we propose a new method to obtain the complete spectra of subKautz sK(d,2) and cyclic Kautz CK(d,3) digraphs, for all d=3, through the Hoffman–McAndrew polynomial and regular partitions. This approach can be useful to find the spectra of other families of digraphs with high regularity.Postprint (author's final draft

    The degree/diameter problem in maximal planar bipartite graphs

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    The (¿;D) (degree/diameter) problem consists of nding the largest possible number of vertices n among all the graphs with maximum degree ¿ and diameter D. We consider the (¿;D) problem for maximal planar bipartite graphs, that are simple planar graphs in which every face is a quadrangle. We obtain that for the (¿; 2) problem, the number of vertices is n = ¿+2; and for the (¿; 3) problem, n = 3¿¿1 if ¿ is odd and n = 3¿ ¿ 2 if ¿ is even. Then, we study the general case (¿;D) and obtain that an upper bound on n is approximately 3(2D + 1)(¿ ¿ 2)¿D=2¿ and another one is C(¿ ¿ 2)¿D=2¿ if ¿ D and C is a sufficiently large constant. Our upper bound improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on n for maximal planar bipartite graphs, which is approximately (¿ ¿ 2)k if D = 2k, and 3(¿ ¿ 3)k if D = 2k + 1, for ¿ and D sufficiently large in both cases.Postprint (published version

    The degree/diameter problem in maximal planar bipartite graphs

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    The (Δ,D)(Δ,D) (degree/diameter) problem consists of finding the largest possible number of vertices nn among all the graphs with maximum degree ΔΔ and diameter DD. We consider the (Δ,D)(Δ,D) problem for maximal planar bipartite graphs, that is, simple planar graphs in which every face is a quadrangle. We obtain that for the (Δ,2)(Δ,2) problem, the number of vertices is n=Δ+2n=Δ+2; and for the (Δ,3)(Δ,3) problem, n=3Δ−1n=3Δ−1 if ΔΔ is odd and n=3Δ−2n=3Δ−2 if ΔΔ is even. Then, we prove that, for the general case of the (Δ,D)(Δ,D) problem, an upper bound on nn is approximately 3(2D+1)(Δ−2)⌊D/2⌋3(2D+1)(Δ−2)⌊D/2⌋, and another one is C(Δ−2)⌊D/2⌋C(Δ−2)⌊D/2⌋ if Δ≥DΔ≥D and CC is a sufficiently large constant. Our upper bounds improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on nn for maximal planar bipartite graphs, which is approximately (Δ−2)k(Δ−2)k if D=2kD=2k, and 3(Δ−3)k3(Δ−3)k if D=2k+1D=2k+1, for ΔΔ and DD sufficiently large in both cases.Peer ReviewedPostprint (published version

    Estudi i disseny de grans xarxes d'interconnexió: modularitat i comunicació

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    Normalment les grans xarxes d'interconnexió o de comunicacions estan dissenyades utilitzant tècniques de la teoria de grafs. Aquest treball presenta algunes contribucions a aquest tema. Concretament, presentem dues noves operacions: el "producte Jeràrquic" de grafs i el "producte Manhattan" de digrafs. El primer és una generalització del producte cartesià de grafs i ens permet construir algunes famílies amb un alt grau de jerarquia, com l'arbre binomial, que és una estructura de dades molt utilitzada en algorísmica. El segon dóna lloc a les conegudes Manhattan Street Networks, les quals han estat extensament estudiades i utilitzades per modelar algunes classes de xarxes òptiques. En el nostre treball, definim formalment i analitzem el cas multidimensional d'aquestes xarxes. Estudiem algunes propietats dels grafs o digrafs obtinguts mitjançant les dues operacions esmentades, especialment: els paràmetres estructurals (les propietats de l'operació, els subdigrafs induïts, la distribució de graus i l'estructura de digraf línia), els paràmetres mètrics (el diàmetre, el radi i la distància mitjana), la simetria (els grups d'automorfismes i els digrafs de Cayley), l'estructura de cicles (els cicles hamiltonians i la descomposició en cicles hamiltonians arc-disjunts) i les propietats espectrals (els valors i vectors propis). En el darrer cas, hem trobat, per exemple, que la família dels arbres binomials tenen tots els seus valors propis diferents, "omplint" tota la recta real. A més a més, mostrem la relació del seu conjunt de vectors propis amb els polinomis de Txebishev de segona espècie. També hem estudiat alguns protocols de comunicació, com els enrutaments locals i els algorismes de difusió. Finalment, presentem alguns models deterministes (com les xarxes Sierpinski i d'altres), els quals presenten algunes propietats pròpies de les xarxes complexes reals (com, per exemple, Internet).Large interconnection or communication networks are usually designed and studied by using techniques from graph theory. This work presents some contributions to this subject. With this aim, two new operations are proposed: the "hierarchical product" of graphs and the "Manhattan product" of digraphs. The former can be seen as a generalization of the Cartesian product of graphs and allows us to construct some interesting families with a high degree of hierarchy, such as the well-know binomial tree, which is a data structure very used in the context of computer science. The latter yields, in particular, the known topologies of Manhattan Street Networks, which has been widely studied and used for modelling some classes of light-wave networks. In this thesis, a multidimensional approach is analyzed. Several properties of the graphs or digraphs obtained by both operations are dealt with, but special attention is paid to the study of their structural parameters (operation properties, induced subdigraphs, degree distribution and line digraph structure), metric parameters (diameter, radius and mean distance), symmetry (automorphism groups and Cayley digraphs), cycle structure (Hamilton cycles and arc-disjoint Hamiltonian decomposition) and spectral properties (eigenvalues and eigenvectors). For instance, with respect to the last issue, it is shown that some families of hypertrees have spectra with all different eigenvalues "filling up" all the real line. Moreover, we show the relationship between its eigenvector set and Chebyshev polynomials of the second kind. Also some protocols of communication, such as local routing and broadcasting algorithms, are addressed. Finally, some deterministic models (Sierpinsky networks and others) having similar properties as some real complex networks, such as the Internet, are presented

    Sequence mixed graphs

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    A mixed graph can be seen as a type of digraph containing some edges (or two opposite arcs). Here we introduce the concept of sequence mixed graphs, which is a generalization of both sequence graphs and literated line digraphs. These structures are proven to be useful in the problem of constructing dense graphs or digraphs, and this is related to the degree/diameter problem. Thus, our generalized approach gives rise to graphs that have also good ratio order/diameter. Moreover, we propose a general method for obtaining a sequence mixed diagraph by identifying some vertices of certain iterated line digraph. As a consequence, some results about distance-related parameters (mainly, the diameter and the average distance) of sequence mixed graphs are presented.Postprint (author's final draft

    Sufficient conditions for a digraph to admit a (1,=l)-identifying code

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    A (1, = `)-identifying code in a digraph D is a subset C of vertices of D such that all distinct subsets of vertices of cardinality at most ` have distinct closed in-neighbourhoods within C. In this paper, we give some sufficient conditions for a digraph of minimum in-degree d - = 1 to admit a (1, = `)- identifying code for ` ¿ {d -, d- + 1}. As a corollary, we obtain the result by Laihonen that states that a graph of minimum degree d = 2 and girth at least 7 admits a (1, = d)-identifying code. Moreover, we prove that every 1-in-regular digraph has a (1, = 2)-identifying code if and only if the girth of the digraph is at least 5. We also characterize all the 2-in-regular digraphs admitting a (1, = `)-identifying code for ` ¿ {2, 3}.Peer ReviewedPostprint (author's final draft

    Moore mixed graphs from Cayley graphs

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    A Moore (r, z, k)-mixed graph G has every vertex with undirected degree r, directed in- and outdegree z, diameter k, and number of vertices (or order) attaining the corresponding Moore bound M(r, z, k) for mixed graphs. When the order of G is close to M(r, z, k) vertices, we refer to it as an almost Moore graph. The first part of this paper is a survey about known Moore (and almost Moore) general mixed graphs that turn out to be Cayley graphs. Then, in the second part of the paper, we give new results on the bipartite case. First, we show that Moore bipartite mixed graphs with diameter three are distance-regular, and their spectra are fully characterized. In particular, an infinity family of Moore bipartite (1, z, 3)-mixed graphs is presented, which are Cayley graphs of semidirect products of groups. Our study is based on the line digraph technique, and on some results about when the line digraph of a Cayley digraph is again a Cayley digraph.This research has been partially supported by AGAUR from the Catalan Government under project 2021SGR00434 and MICINN from the Spanish Government under project PID2020-115442RBI00.Peer ReviewedPostprint (published version

    New cyclic Kautz digraphs with optimal diameter

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    We obtain a new family of digraphs with minimal diameter, that is, given the number of vertices and out-degree, there is no other digraph with a smaller diameter. This new family of digraphs are called `modified cyclic digraphs' M C K ( d , l ) , and it is derived from the Kautz digraphs K ( d , l ) and from the so-called cyclic Kautz digraphs C K ( d , l ) . The cyclic Kautz digraphs C K ( d , l ) were defined as the digraphs whose vertices are labeled by all possible sequences a 1 … a l of length l , such that each character a i is chosen from an alphabet of d + 1 distinct symbols, where the consecutive characters in the sequence are different (as in Kautz digraphs), and also requiring that a 1 ¿ a l . Their arcs are between vertices a 1 a 2 … a l and a 2 … a l a l + 1 , with a 1 ¿ a l and a 2 ¿ a l + 1 . Since C K ( d , l ) do not have minimal diameter for their number of vertices, we construct the modified cyclic Kautz digraphs to obtain the same diameter as in the Kautz digraphs, and we also show that M C K ( d , l ) are d -out-regular. Moreover, for t = 1 , we compute the number of vertices of the iterated line digraphs L t ( C K ( d , l ) ) .The research of C. Dalfó has been partially supported by AGAUR from the Catalan Government under project 2017SGR1087, and by MICINN from the Spanish Government under projects MTM2017-83271-R and PGC2018-095471-B-I00. The research of C. Huemer was supported by PID2019-104129GB-I00/AEI/ 10.13039/501100011033 and Gen. Cat. DGR 2017SGR1336. This work has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 734922 .Postprint (author's final draft
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