Showing papers on "Voltage graph published in 1975"

Finding All the Elementary Circuits of a Directed Graph

Abstract: An algorithm is presented which finds all the elementary circuits of a directed graph in time bounded by $O((n + e)(c + 1))$ and space bounded by $O(n + e)$, where there are n vertices, e edges and c elementary circuits in the graph. The algorithm resembles algorithms by Tiernan and Tarjan, but is faster because it considers each edge at most twice between any one circuit and the next in the output sequence.

Design of cellular production systems A graph-theoretic approach

Abstract: SUMMARY A method is presented for forming machine cells while implementing group technology. Information derived from the route cards of the components is analysed and the situation is represented in the form of a graph whose vertices correspond to the machines and whose edges represent the relationships created between the machines by the components using them. For finding the cells a graph partitioning approach is suggested and developed. Results obtained from testing the method using actual data are also presented.

Some Graph-Theoretical Aspects of Simple Ring Current Calculations on Conjugated Systems

Abstract: In this paper we examine the simple theory of $\pi $-electron ring currents in conjugated systems (devised originally by London (1937) and extended by Pople (1958) and McWeeny (1958)), with particular reference to the topological or graph-theoretical aspects of it (all the necessary graph-theoretical ideas and terminology are explained in the text). There is a close correspondence between the adjacency matrix of the graph representing the $\sigma $-bond skeleton of the carbon atoms comprising a given conjugated system, and the secular equations which arise in the theory (a relation now well known to be common to all formalisms based on Huckel 'topological' molecular orbitals), but in addition we here emphasize that several other graph-theoretical ideas-notably those concerning circuits and spanning trees-specifically underlie the ring current concept. In this connexion, the question of whether any given molecular graph is semi-Hamiltonian or non-Hamiltonian is of prime importance, and it is pointed out that a unitary transformation originally proposed by McWeeny applies to semi-Hamiltonian molecular graphs, whereas one recently devised by Gayoso & Boucekkine can be applied to any simple, connected graph-as also can an explicit ring current formula (based on the London-McWeeny theory) just published by the present author. These ideas are illustrated by some simple numerical calculations, and an example is given of a conjugated system (decacyclene) whose molecular graph is apparently non-Hamiltonian. It is emphasized that although much graph theory is inherent in the ring current concept, the ring current index itself is not a completely topological quantity-even when a purely topological wavefunction (such as the simple Huckel one) has been used to calculate it.

The graph of the walras correspondence

Abstract: THE WALRAS CORRESPONDENCE associates the set equilibrium price vectors with an economy. The purpose of this paper is to study some topological properties of the graph of the Walras correspondence such as connectedness and contractibility. This is done once the graph is given the structure of a bundle. The mathematical notations used in this paper are given in Section 2. The bundle structure of the graph is proved in Section 3. The contractibility of the graph is then a straightforward result proved in Section 4. Finally, variable demand functions are introduced in Section 5 and connectivity is then proved.

Graphs from Projective Planes

Abstract: The orthogonality relation among subspaces of a finite vector space is studied here by means of the corresponding graph. In the case we consider, this graph has some highly symmetric induced subgraphs. We find three infinite families of graphs of girth 3, and two infinite families of graphs of girth 5, whose automorphism groups are transitive on ordered pairs of adjacent points. Special cases are the Petersen graph, a 28-point cubic 3-transitive graph of girth 7 due to H. S. M. Coxeter, and a 36-point quintic 2-transitive graph of girth 5. The algebraic representations obtained for these graphs afford easy computation of their properties, and show that their automorphism groups are respectively the projective orthogonal groups PC/3 (5), P6 3 (7), and a semidirect product P~3 (9)# Z z. Our study is in the spirit of Coxeter's paper 'Self-dual configurations and regular graphs' [4]. In that paper, many interesting graphs were obtained as the Levi graphs of geometric configurations. A configuration (m c, ha) is a set ofm points and n lines in a plane, with dpoints on each line and c lines through each point. The configuration is self-dual if it has a duality (incidence-preserving bijection interchanging points and lines). The Levi graph is the red-blue bipartite graph joining each point (red vertex) to each line (blue vertex) incident to it in the configuration. This graph always has even girth. A polarity n is a duality such that 7~ 2 is the identity. We start from a configuration self-dual via a polarity n. Instead of the Levi graph, we use a 'polarity graph' whose vertices are the points X, Y,... of the configuration, and where X is adjacent to Y in the graph ifX-~ Yand X is on n(Y), the polar line of Y. Our configuration consists of the points and lines of the finite projective plane PG(2, q), where q is a prime power. The points and lines may be identified with the I-dimensional and 2-dimensional subspaces of the 3-dimensional vector space over the finite field F= GF(q). We let n (X) be the orthogonal complement of X. The resulting polarity graph is denoted by G(q).

How to color a graph

Abstract: This paper attempts to give a short survey of graph coloring problems. The first two sections deal with edge colorings and node colorings of graphs. In section 4 some generalizations to hyper-graph coloring are described. Finally section 5 is devoted to the problem of balancing the colorings. For notations and definitions we follow C. Berge [4]. The multigraphs considered here have no loops.

On finding maximum compatibles

Abstract: A description of a way of finding the maximum compatibles (maximal complete subgraphs of an undirected graph) is presented. Only a few operations are needed and at each step the graph under consideration is reduced.

Applications of graph theory to matrix theory

Abstract: Let A l Ak be n x n matrices over a commutative ring R with identity. Graph theoretic methods are established to compute the standard polynomial [A , ... I Ak]. It is proved that if k < 2n 2, and if the characteristic of R either is zero or does not divide 4I(V2 n) 2, where I denotes the greatest integer function, then there exist n x n skew-symmetric matrices A 1 . . . , Ak such that [A 1, . . . AkI AO.

Necessary and sufficient condition for a graph to be three-terminal series-parallel

Abstract: A "three-terminal series-parallel graph" is defined to be a three-terminal graph which is constructed by means of repeating only specified series and parallel connections. This definition is based upon our previous work about transformerless interconnections of two-port networks. Some properties of the graph are shown under the foregoing definition and, especially, a theorem is given stating that a three-terminal graph is three-terminal series-parallel if and only if neither of two certain graphs can be obtained from it by opening or shorting some edges.

The graph of the chromial of a graph

Abstract: The chromial or chromatic polynomial of a finite graph G is a polynomial P(G,λ) in a variable λ with the following property: the value of P(G,λ) when λ is a positive integer is the number of ways of colouring G in λ colours. In this paper various properties of the chromial, considered as a function of a real variable λ, are discussed.

A Graph Problem of Structural Design

Abstract: Without repeating an old argument [1, 2, 3, 4] in great detail, the state-of-the art of optimization in structural design is such that one of the next steps must be the automation of the selection of the structural configuration (connectivity). While there are both topological and geometrical considerations to be dealt with here, algorithms are beginning to appear (cf., e.g. [5]) which will take a given configuration and move the joints to produce an optimal or at least an improved design. The question which seems to have received the least study of all is how the pieces of a structure should be jointed together to produce an optimal configuration. For skeletal structures this is, of course, a graph problem.

The Entire Graph of a Bridgeless Connected Plane Graph is Panconnected

Abstract: Recently A. M. Hobbs and J. Mitchem [7] proved that the entire graph of a bridge-less connected plane graph is Hamiltonian. In this paper we strengthen this result substantially by showing that entire graphs of such plane graphs are panconnected. (Between each pair of distinct vertices in a panconnected graph there exist paths of all lengths greater than or equal to the distance between the vertices.) This fits a pattern which indicates that Hamiltonian-connected graphs seem to have paths of \" many \" lengths between each pair of distinct points [1,2, 3]. The graphs we consider will be undirected, finite, and have no loops or multiple edges. A plane graph is a graph already embedded in the plane. If G is a plane graph, V(G), E(G) and F(G) denote the sets of its vertices, edges and faces, respectively. Two distinct vertices (edges, faces) of G are adjacent if they share a common edge (vertex, edge). A vertex and an edge, a vertex and a face, or an edge and a face, are adjacent if they are incident (in the obvious sense). The entire graph of G, denoted e(G), is the graph with vertex set V(G) u E(G) u F(G), with two vertices of e(G) adjacent if and only if they are adjacent in G. Hamiltonian and Eulerian properties of entire graphs were first discussed by J. Mitchem in [8]. By P(l) (respectively C(/)) we will mean a path (respectively circuit, i.e., cycle) with /vertices. A path with (distinct) vertices v u v 2 ,..., v t and edges e u e 2 ,..., e x-\\ will be written (y l5 v 2 ,..., v t) or [v lt e lt v 2> e 2 ,..., e,_ l5 v t ], while (v it v 2> ..., v t , Vi) denotes a circuit with the same vertices. If P = (v 2 , v 3 ,..., u,), then (v u P, v l+ x) denotes the path (v l9 v 2 ,..., v h v l+l) (if it exists). If u ^ v, d G (u, v) will denote the distance between u and v in G, while P,(w, V) will denote a path between u and v containing / vertices. If P,(w, v) exists for all u ^ v in V(G) and for all /, d G (u, v) < I ^ | V(G)\\, then G is called panconnected. …

Generation of all possible trees of a graph in independent groups

Abstract: A special graph, the PS graph, is introduced and an algorithm is developed to generate all trees of such graphs. It is proved that for any given graph, G there exists certain number of PS graphs, obtained from G , such that the collection of all trees of all such PS graph span all trees of G with no duplication. In addition to a number of properties of PS graphs indicated, the procedure seems very useful for topological analysis and design of networks, or any other types of systems that can be represented by a linear graph.

Exklusive Graphen und Hamiltonche Graphenn-ten Grades II

Abstract: The graphs considered are finite and undirected, loops do not occur. An induced subgraphI of a graphX is called animitation ofX, if In the first chapter some theorems concerning exclusive graphs and Euler graphs are stated. Chapters 2 deals withHG n′ s and bipartite graphs. In chapters 3 a useful concept—theX-graph of anHG n—is defined; in this paper it is the conceptual connection between exclusive graphs andHG n′ s, since a graphG is anHG n, if all itsX-graphs are exlusive. Furthermore, some theorems onX-graphs are proved. Chapter 4 contains the quintessence of the paper: If we want to construct a newHG n F from anotherHG n G, we can consider certain properties of theX-graphs ofG to decide whetherF is also anHG n.

Finding the Maximal Incidence Matrix of a Large Graph.

Abstract: : The paper deals with the computation of two canonical representations of a graph. A computer program is presented which searches for 'the maximal incidence matrix' of a large connected graph without multiple edges or self-loops. The use of appropriate algorithms and data structures is discussed.

On the Connectivities of the Generalized Line, Middle and Total Graph

Abstract: We investigate the generalized line graph, total graph and middle graph of the given graphs. First, replace a given graph G with a multigraph M having the same point set as G, then it constructs the total graph T(G) from G. Two suitable subgraphs Mx(G), Lx(G), of Tx(G) are considered as extensions of middle graph M(G) and line graph L(G) respectively.We also consider the connectivities of graphs which are obtained as a special case of Lx(G), Mx(G) and Tx(G). Then we get some extensions of established theorems.

The open mapping and closed graph theorem for embeddable topological semigroups

Abstract: Some extensions of the open mapping and closed graph theorem are proved for certain classes of commutative topological semigroups, namely those embeddable as open subsets of topological groups. Preliminary results of independent interest include investigations of properties which “lift” from embeddable semigroups to the groups in which they are embedded, and from semigroup homomorphisms to homomorphisms of the groups.