DOCSLIB.ORG

Generalized Data Flow Graphs : Theory and Applications

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Generalized data flow graphs : theory and applications Citation for published version (APA): Jong, de, G. G. (1993). Generalized data flow graphs : theory and applications. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR402436 DOI: 10.6100/IR402436 Document status and date: Published: 01/01/1993 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at: [email protected] providing details and we will investigate your claim. Download date: 08. Oct. 2021 Generalized data flow graphs theory and applications c ____ ,._ _____ _ Gjalt Gerrit de Jong Generalized data flow graphs theory and applications Generalized data flow graphs theory and applications PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de Rector Magnificus, prof. dr. J.H. van Lint, voor een commissie aangewezen door het College van Dekanen in het open­ baar te verdedigen op vrijdag 8 oktober 1993 om 16.00 uur door Gjalt Gerrit de Jong geboren te Bierum Oit proefschrift is goedgekeurd door de promotoren prof. Dr. -Ing. J.A.G. Jess en prof. dr. ir. Th. Krol en door de copromotor dr. ir. J.T.J. van Eijndhoven © Copyright 1993 Gjalt Gerrit de Jong CIP-GEGEVENS KONINKLIJKE BIBLIOTHEEK, DEN HAAG Jong, Gjalt Gerrit de Generalized data flow graphs : theory and applications I Gjalt Gerrit de Jong. - [S.I. : s.n.]. - Fig., tab. Thesis Eindhoven. - With index, ref. - With summary in Dutch ISBN 90-9006384-6 NUGI 852/ 832 Subject headings: programming theory I programming verification I formal languages Abstract The area of application for the concepts defined and the theorems and properties proved in this thesis, is the automatic generation of integrated circuits. Starting from a high level description of the specification of the function of a system, high level synthesis results in a network graph, i.e. a network or a description at the structural (register transfer) level, and a control graph, defining which modules are active at what (time) instance. The network graph is the main result of the alloca­ tion phase of high level synthesis, whereas the control graph is the main result of the scheduling phase. The high level description typically is a program or algo­ rithm describing the function of the system to be designed. The concept of data flow graphs is a well-suited formalism to start synthesis from, since it contains all the necessary details to do synthesis, without the bother by a specific syntax or the syntactic sugar of any language. It is therefore a general formalism. A data flow graph imposes also no other restriction on the execution of the system than those prescribed by the data dependencies in the high level description. In this thesis, the concept of data flow graphs is extended with so-called multi­ destination and multi-origin edges, i.e. hyper edges. Data flow graphs convention­ ally have only edges with one origin and one destination. The introduction of multi-destination edges does not lead to any additional expressiveness, but the multi-origin feature does. The multi-origin edges lead to the notion of 'choice'. In this way, networks are a special type of these generalized data flow graphs, namely those in which all nodes are defined by various kinds of hardware modules instead of by some abslract type of behavior. The multi-destination and multi-origin edges Vi ABSTRAC'T in the data flow graph are just nets in a network. Also it is shown that the control graph is an abstraction of the data flow graph under consideration. Therefore, high level synthesis is just a graph transformation on the data flow graph level. Scheduling turns out to be a partitioning of the flow graph, which is implied by the introduction of sequence edges. Sequence edges have in principle the same func­ tion as the normal data edges, namely to denote a precedence relation. Another important task of high level synthesis is to minimize the number of multiplexors and demultiplexors. These (de)multiplexors are introduced by scheduling (in which they normally are not mentioned explicitly at all), and by the control state­ ments, like if and while, in the specification. Scheduling also is just a graph trans­ formation that can be proved to be equivalence preserving. With the concepts pre­ sented here, some requirements are stated under which synthesis results in a prop­ erly and equivalently behaving network, which is only restricted in the number of execution sequences. These requirements turn out to be less restrictive than those of existing synthesis tools. This gives the opportunity to find better solutions. Nonnal data flow graphs may be viewed as marked Petri nets, while choice is included with the generalized data flow graph presented here, leading to possible non-determinism. In principle the notion of conflict is also added. Most results assume a conflict-free flow graph, since conflict is not used for the type of applica­ tion of high level synthesis. However, conflict is considered in one of the main results, namely the reduction technique to compute as little as possible of the reachable state space of a data flow graph. This technique is called anticipation. It is e.g. proved that a choice-free graph is deterministic, and that only one state sequence of the reachable state space is necessary to extract the behavior of the flow graph. When the data flow graph is not choice-free, only a few execution sequences are needed. Since conflict in normal, i.e. uninterpreted, Petri nets and choice in data flow graphs are each other's dual, the same result is valid for Petri nets. Several semantics are defined for the generalized data flow graphs: an operational semantics and two denotational semantics. One denotational semantics defines only the (input-output) behavior of a flow graph, while the other also defines the different execution sequences, i.e. the internal behavior, too. Another presented denotational semantics is based on a process algebra. All these semantics are proved to be equivalent, and they reduce to the conventional semantics as defined for normal, i.e. choice-free, data flow graphs. With these semantics, some prop­ erty preserving transformations are defined, mainly of use for hierarchical expan­ sion of nodes as well as for abstraction. ABSTRACT vii Furthermore, some examples are given to show the absence of any other restriction in the data flow graph concept, apart from the data dependencies. These examples show the power of the anticipation strategy, too, even for Petri nets, and illustrate some of the other stated theorems. It is also shown how data flow graphs can be used for specifications which at first sight violate the data flow principle, as is the case for some constructs commonly used in parallel programs and in control domi­ nated specifications. Also a technique is described which can be used to verify requirements that a sys­ tem should satisfy, and which are not part of the functional description. Examples of such additional requirements are the freedom of deadlock and the absence of starvation for a set of cooperating flow graphs. The technique is model checking of temporal logic formulae. The use of temporal logic allows for a large variety of properties to be checked, and is not restricted to the classical ones such as liveness and safeness in Petri nets. An extension to this method of model checking is pre­ sented, in which constraints are suggested under which the system under consider­ ation will satisfy certain additional requirements, if possible. These constraints are of the form of additional sequence edges, so that only the correctly behaving sub­ set of the state space is reachable. Samenvatti11g Het toepassingsgebied van de concepten die in dit proefschrift worden gedefinieerd, en van de theorema's en eigenschappen die worden bewezen, is het automatisch genereren van gefutegreerde schakelingen. Uitgaande van een hoog niveau beschrijving van de specificatie van de functie van een systeem, leidt hoog niveau syntbese tot een netwerk graaf, dat wil zeggen een beschrijving op structuur-, of register-transfer, niveau, en een controle-graaf, die definieert welke modulen actief zijn op welk moment.
Recommended publications
  • Descriptive Graph Combinatorics

    Descriptive Graph Combinatorics

    Descriptive Graph Combinatorics Alexander S. Kechris and Andrew S. Marks (Preliminary version; November 3, 2018) Introduction In this article we survey the emerging field of descriptive graph combina- torics. This area has developed in the last two decades or so at the interface of descriptive set theory and graph theory, and it has interesting connections with other areas such as ergodic theory and probability theory. Our object of study is the theory of definable graphs, usually Borel or an- alytic graphs on Polish spaces. We investigate how combinatorial concepts, such as colorings and matchings, behave under definability constraints, i.e., when they are required to be definable or perhaps well-behaved in the topo- logical or measure theoretic sense. To illustrate the new phenomena that can arise in the definable context, consider for example colorings of graphs. As usual a Y -coloring of a graph G = (X; G), where X is the set of vertices and G ⊆ X2 the edge relation, is a map c: X ! Y such that xGy =) c(x) 6= c(y). An elementary result in graph theory asserts that any acyclic graph admits a 2-coloring (i.e., a coloring as above with jY j = 2). On the other hand, consider a Borel graph G = (X; G), where X is a Polish space and G is Borel (in X2). A Borel coloring of G is a coloring c: X ! Y as above with Y a Polish space and c a Borel map. In contrast to the above basic fact, there are acyclic Borel graphs G which admit no Borel countable coloring (i.e., with jY j ≤ @0); see Example 3.14.
  • Calculating Graph Algorithms for Dominance and Shortest Path*

    Calculating Graph Algorithms for Dominance and Shortest Path*

    Calculating Graph Algorithms for Dominance and Shortest Path⋆ Ilya Sergey1, Jan Midtgaard2, and Dave Clarke1 1 KU Leuven, Belgium {first.last}@cs.kuleuven.be 2 Aarhus University, Denmark [email protected] Abstract. We calculate two iterative, polynomial-time graph algorithms from the literature: a dominance algorithm and an algorithm for the single-source shortest path problem. Both algorithms are calculated di- rectly from the definition of the properties by fixed-point fusion of (1) a least fixed point expressing all finite paths through a directed graph and (2) Galois connections that capture dominance and path length. The approach illustrates that reasoning in the style of fixed-point calculus extends gracefully to the domain of graph algorithms. We thereby bridge common practice from the school of program calculation with common practice from the school of static program analysis, and build a novel view on iterative graph algorithms as instances of abstract interpretation. Keywords: graph algorithms, dominance, shortest path algorithm, fixed-point fusion, fixed-point calculus, Galois connections 1 Introduction Calculating an implementation from a specification is central to two active sub- fields of theoretical computer science, namely the calculational approach to pro- gram development [1,12,13] and the calculational approach to abstract interpre- tation [18, 22, 33, 34]. The advantage of both approaches is clear: the resulting implementations are provably correct by construction. Whereas the former is a general approach to program development, the latter approach is mainly used for developing provably sound static analyses (with notable exceptions [19,25]). Both approaches are anchored in some of the same discrete mathematical structures, namely partial orders, complete lattices, fixed points and Galois connections.
  • The Characteristic Polynomial of Multi-Rooted

    The Characteristic Polynomial of Multi-Rooted

    THE CHARACTERISTIC POLYNOMIAL OF MULTI-ROOTED, DIRECTED TREES LAUREN EATON Mathematics 8 May 2008 Independent Research Project: Candidate for Departmental Honors Graduation: 17 May 2008 THE CHARACTERISTIC POLYNOMIAL OF MULTI-ROOED, DIRECTED TREES LAUREN EATON Abstract. We define the characteristic polynomial for single-rooted trees and begin with a theorem about this polynomial, derived from the known contrac- tion/deletion formula. We expand our scope to include multi-rooted, directed trees. Introducing the concept of a star, we prove two theorems which allow us to evaluate the polynomials of these trees in terms of the stars that comprise them. Finally, we derive and prove a general formula for the characteristic polynomial of multi-rooted, directed trees. Introduction Introduced by Gary Gordon and Elizabeth McMahon in [3] the greedoid char- acteristic polynomial is a generalization of the Tutte polynomial, which is a two- variable invariant which helps describe the structure of a single-rooted graph. Given a multi-rooted, directed tree, one must conduct an often tedious recursive calcula- tion in order to obtain the polynomial. In this paper, we work towards defining a general formula for the characteristic polynomial that requires minimal calculation and is entirely in terms of the vertices. This paper is organized as follows: In Section 1, we start with a focus on single- rooted trees and an adaptation of the characteristic polynomial. In Section 2, we look at the more complex multi-rooted, directed trees. We will find that when deriving the characteristic polynomial of these complicated trees, one can consider them in simplified terms. We will explore the concept of a sink and see how the polynomial is affected by its presence.
  • Rooted Graph Minors and Reducibility of Graph Polynomials Arxiv

    Rooted Graph Minors and Reducibility of Graph Polynomials Arxiv

    Rooted Graph Minors and Reducibility of Graph Polynomials by Benjamin Richard Moore B.A., Thompson Rivers University, 2015 Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of Masters of Science in the Department of Mathematics and Statistics Faculty of Science c Benjamin Richard Moore 2017 SIMON FRASER UNIVERSITY Spring 2017 arXiv:1704.04701v1 [math.CO] 15 Apr 2017 All rights reserved. However, in accordance with the Copyright Act of Canada, this work may be reproduced without authorization under the conditions for “Fair Dealing.” Therefore, limited reproduction of this work for the purposes of private study, research, criticism, review and news reporting is likely to be in accordance with the law, particularly if cited appropriately. Approval Name: Benjamin Richard Moore Degree: Masters of Science (Mathematics) Title: Rooted Graph Minors and Reducibility of Graph Poly- nomials Examining Committee: Dr. Ladislav Stacho (chair) Associate Professor Dr. Karen Yeats Senior Supervisor Associate Professor Dr. Luis Goddyn Supervisor Professor Dr. Bojan Mohar Internal Examiner Professor Date Defended: April 6, 2017 ii Abstract In 2009, Brown gave a set of conditions which when satisfied imply that a Feynman integral evaluates to a multiple zeta value. One of these conditions is called reducibility, which loosely says there is an order of integration for the Feynman integral for which Brown’s techniques will succeed. Reducibility can be abstracted away from the Feynman integral to just being a condition on two polynomials, the first and second Symanzik polynomials. The first Symanzik polynomial is defined from the spanning trees of a graph, and the second Symanzik polynomial is defined from both spanning forests of a graph and some edge and vertex weights, called external momenta and masses.
  • Large, Lengthy Graphs Look Locally Like Lines

    Large, Lengthy Graphs Look Locally Like Lines

    Large, Lengthy Graphs Look Locally Like Lines Itai Benjamini and Tom Hutchcroft May 9, 2019 Abstract We apply the theory of unimodular random rooted graphs to study the metric geometry of large, finite, bounded degree graphs whose diameter is proportional to their volume. We prove that for a positive proportion of the vertices of such a graph, there exists a mesoscopic scale on which the graph looks like R in the sense that the rescaled ball is close to a line segment in the Gromov-Hausdorff metric. 1 Introduction The aim of this modest note is to prove that large graphs with diameter proportional to their volume must `look like R' from the perspective of a positive proportion of their vertices, after some loc rescaling that may depend on the choice of vertex. We write dGH for the local Gromov-Hausdorff metric, a measure of similarity between locally compact pointed metric spaces that we define in detail in Section 1.1. Theorem 1.1. Let (Gn)n≥1 = ((Vn;En))n≥1 be a sequence of finite, connected graphs with jVnj ! 1, and suppose that there exists a constant C < 1 such that jVnj ≤ C diam(Gn) for every n ≥ 1. Suppose furthermore that the set of degree distributions of the graphs Gn are uniformly integrable. −1 Then there exists a sequence of subsets An ⊆ Vn with lim infn!1 jAnj=jVnj ≥ C such that loc lim sup inf dGH Vn; "dGn ; v ; ; d ; 0 = 0; n!1 ">0 R R v2An arXiv:1905.00316v3 [math.MG] 8 Nov 2020 where we write dGn for the graph metric on Gn and dR(x; y) = jx − yj for the usual metric on R.
  • Two Proofs of Cayley's Theorem

    Two Proofs of Cayley's Theorem

    TWO PROOFS OF CAYLEY'S THEOREM TITU ANDREESCU AND COSMIN POHOATA Abstract. We present two proofs of the celebrated Cayley theorem that the number of spanning trees of a complete graph on n vertices is nn−2. In this expository note we present two proofs of Cayley's theorem that are not as popular as they deserve to be. To set up the story we revisit first some terminology. By a graph G we mean a pair (V (G);E(G)), where V (G) is a set of points (or vertices) and E(G) is a subset of V (G) × V (G). Visually, each element from the set E(G) can be represented as a curve connecting the two corresponding points, thus we will call these the edges of graph G. Note that with the above definition, graphs cannot have multiple edges between two given vertices u and v, but they can have loops (i.e. edges in E(G) of the form (w; w) for w 2 V (G)). However, in this paper we won't allow such pairs. Such simple graphs can be directed or undirected depending on whether the pairs in E(G) ⊂ V (G) × V (G) are ordered or not. A path is a graph P with vertex set V (P ) = fv1; : : : ; vng and edge set E(P ) = f(vi; vi+1) : 1 ≤ i ≤ n − 1g. The number n will be refered to as the length of the path. A cycle (of length n) is a graph C of the form C = P [ (vn; v1). Or in other words, a cycle of length n is a graph obtained from a path of length n by adding an edge between its endpoints.
  • Locally Interacting Diffusions As Space-Time Markov Random Fields 3

    Locally Interacting Diffusions As Space-Time Markov Random Fields 3

    LOCALLY INTERACTING DIFFUSIONS AS SPACE-TIME MARKOV RANDOM FIELDS DANIEL LACKER, KAVITA RAMANAN, AND RUOYU WU Abstract. We consider a countable system of interacting (possibly non-Markovian) stochastic differential equations driven by independent Brownian motions and indexed by the vertices of a locally finite graph G = (V,E). The drift of the process at each vertex is influenced by the states of that vertex and its neighbors, and the diffusion coefficient depends on the state of only that vertex. Such processes arise in a variety of applications including statistical physics, neuroscience, engineering and math finance. Under general conditions on the coefficients, we show that if the initial conditions form a second-order Markov random field on d-dimensional Euclidean space, then at any positive time, the collection of histories of the processes at dif- ferent vertices forms a second-order Markov random field on path space. We also establish a bijection between (second-order) Gibbs measures on (Rd)V (with finite second moments) and a set of space-time (second-order) Gibbs measures on path space, corresponding respectively to the initial law and the law of the solution to the stochastic differential equation. As a corollary, we establish a Gibbs uniqueness property that shows that for infinite graphs the joint distri- bution of the paths is completely determined by the initial condition and the specifications, namely the family of conditional distributions on finite vertex sets given the configuration on the complement. Along the way, we establish various approximation and projection results for Markov random fields on locally finite graphs that may be of independent interest.
  • Balancing Problems in Acyclic Networks*

    Balancing Problems in Acyclic Networks*

    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector DISCRETE APPLIED MATHEMATICS ELSEVIER Discrete Applied Mathematics 49 (1994) 77-93 Balancing problems in acyclic networks* Endre Borosa* b, **, Peter L. Hammerb, Mark E. Hartmannc, Ron Shamird**** n DIMACS, Rutgers University, New Brunswick, NJ 08903, USA ‘RUTCOR. Rutgers Universify, New Brunswick, NJ 08903, USA ‘Department of Operaiions Research, University qf North Carolina, Chapel Hill, NC 27599, USA d Department ofComputer Science, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel-Aviv 69978, Israel Received 15 August 1991; revised 17 March 1992 Abstract A directed acyclic network with nonnegative integer arc lengths is called balanced if any two paths with common endpoints have equal lengths. In the buffer assignment problem such a network is given, and the goal is to balance it by increasing arc lengths by integer amounts (called buffers), so that the sum of the amounts added is minimal. This problem arises in VLSI design, and was recently shown to be polynomial for rooted networks. Here we give simple procedures which solve several generalizations of this problem in strongly polynomial time, using ideas from network flow theory. In particular, we solve a weighted version of the problem, extend the results to nonrooted networks, and allow upper bounds on buffers. We also give a strongly polynomial algorithm for solving the min-max buffer assignment problem, based on a strong proximity result between fractional and integer balanced solutions. Finally, we show that the problem of balancing a network while minimizing the number of arcs with positive buffers is NP-hard.
  • A Characteristic Polynomial for Rooted Graphs and Rooted Digraphs

    A Characteristic Polynomial for Rooted Graphs and Rooted Digraphs

    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 232 (2001) 19–33 www.elsevier.com/locate/disc A characteristic polynomial for rooted graphs and rooted digraphs Gary Gordon ∗, Elizabeth McMahon Mathematics Department, Lafayette College, Easton, PA 18042, USA Received 4 February 1999; revised 27 March 2000; accepted 10 April 2000 Abstract We consider the one-variable characteristic polynomial p(G; ) in two settings. When G is a rooted digraph, we show that this polynomial essentially counts the number of sinks in G. When G is a rooted graph, we give combinatorial interpretations of several coe/cients and the degree of p(G; ). In particular, |p(G;0)| is the number of acyclic orientations of G, while the degree of p(G; ) gives the size of the minimum tree cover (every edge of G is adjacent to some edge of T), and the leadingcoe/cient givesthe number of such covers. Finally, we consider the class of rooted fans in detail; here p(G; ) shows cyclotomic behavior. c 2001 Elsevier Science B.V. All rights reserved. Keywords: Characteristic polynomial; Rooted graph; Rooted digraph; Branching greedoid 1. Introduction Rooted graphs and digraphs are important combinatorial structures that have wide ap- plication, but they have received relatively little attention from the viewpoint of graphic invariants. A fundamental reason for this oversight is that although the Tutte polyno- mial, characteristic polynomial, ÿ-invariant, and other invariants have been well-studied for ordinary graphs and matroids, rooted graphs and digraphs do not have a matroidal rank function.
  • Fast Graph Programs Project Report

    Fast Graph Programs Project Report

    Fast Graph Programs Project Report Graham Campbell, Jack Romo, and Detlef Plump Department of Computer Science University of York, United Kingdom Summer 2018 Contents Prologue 3 1 Introduction4 1.1 Background...................................4 1.2 Project Aims..................................5 2 Automata Simulation6 2.1 Motivation....................................6 2.2 FSAs.......................................6 2.3 DPDAs..................................... 10 3 Efficient Topological Sort 13 3.1 Problem Description.............................. 13 3.2 GP2 Implementation.............................. 14 3.3 Complexity Results............................... 16 4 Efficient Red-Black Trees 18 4.1 Problem Description.............................. 18 4.2 Functional RBTs................................ 18 4.3 GP2 Implementation.............................. 19 4.4 Complexity Results............................... 28 1 of 49 5 Program Refinement 29 5.1 Motivation.................................... 29 5.2 Refinement I................................... 31 5.3 Refinement II.................................. 33 5.4 Refinement III.................................. 34 5.5 Refinement of List Sorting........................... 35 6 Conclusion 38 A Project Tooling 39 A.1 Overview..................................... 39 A.2 GP2I....................................... 40 A.3 GP2 Bench................................... 42 A.4 LaTeX...................................... 44 B Proofs 47 B.1 Complexity of RBT Insertion......................... 47 Bibliography
  • Factorisation of Greedoid Polynomials of Rooted Digraphs Arxiv

    Factorisation of Greedoid Polynomials of Rooted Digraphs Arxiv

    Factorisation of Greedoid Polynomials of Rooted Digraphs Kai Siong Yow∗1, Kerri Morgany2, and Graham Farrz1 1Faculty of Information Technology, Monash University, Clayton, Victoria 3800, Australia 2Deakin University, Geelong, Australia, School of Information Technology, Faculty of Science Engineering & Built Environment May 6, 2019 Abstract Gordon and McMahon defined a two-variable greedoid polynomial f(G; t; z) for any greedoid G. They studied greedoid polynomials for greedoids associated with rooted graphs and rooted digraphs. They proved that greedoid polyno- mials of rooted digraphs have the multiplicative direct sum property. In ad- dition, these polynomials are divisible by 1 + z under certain conditions. We compute the greedoid polynomials for all rooted digraphs up to order six. A greedoid polynomial f(D) of a rooted digraph D of order n GM-factorises if f(D) = f(G) · f(H) such that G and H are rooted digraphs of order at most n and f(G); f(H) =6 1. We study the GM-factorability of greedoid polynomials of rooted digraphs, particularly those that are not divisible by 1 + z. We give some examples and an infinite family of rooted digraphs that are not direct sums but their greedoid polynomials GM-factorise. Keywords: factorisation, greedoid polynomial, greedoid, directed branching gree- doid, rooted digraph, arborescence 1 Introduction arXiv:1809.02924v2 [math.CO] 3 May 2019 Greedoids were introduced by Korte and Lov´aszas collections of sets that generalise matroids [11]. Korte and Lov´aszobserved that the optimality of some \greedy" algo- rithms including breadth-first search could be traced back to an underlying combina- torial structure that satisfies the greedoid, but not the matroid, framework.
  • Chemical Graphs Constructed from Rooted Product and Their Zagreb Indices

    Chemical Graphs Constructed from Rooted Product and Their Zagreb Indices

    MATCH MATCH Commun. Math. Comput. Chem. 70 (2013) 901-919 Communications in Mathematical and in Computer Chemistry ISSN 0340 - 6253 Chemical Graphs Constructed from Rooted Product and Their Zagreb Indices Mahdieh Azari and Ali Iranmanesh* Department of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, P. O. Box: 14115-137, Tehran, Iran [email protected] (Received March 16, 2013) Abstract. Rooted product of n vertex graph H by a sequence of n rooted graphs G1,G2,...,Gn , is the graph obtained by identifying the root vertex of Gi with the i-th vertex of H for all i 1,2,...,n . In this paper, we show how the first and second Zagreb indices of rooted product of graphs are determined from the respective indices of the individual graphs. The first and second Zagreb indices of cluster of graphs, thorn graphs and bridge graphs as three important special cases of rooted product are also determined. Using these formulae, the first and second Zagreb indices of several important classes of chemical graphs will be computed. 1. Introduction In this paper, we consider connected finite graphs without any loops or multiple edges. Let G be such a graph with the vertex set V(G) and the edge set E(G) . For u +V(G) , we denote by NG (u) the set of all neighbors of u in G. Cardinality of the set NG (u) is called the degree of u in G and will be denoted by deg G (u) . We denote by G (u) , the sum of degrees of all neighbors of the vertex u in G, i.e., G (u) Bdeg G (a) .