Frank Harary

A SURVEY OF THE THEORY OF HYPERCUBE GRAPHS

Abstract We present a comprehensive survey of the theory of hypercube graphs. Basic properties related to distance, coloring, domination and genus are reviewed. The properties of the n-cube defined by its subgraphs are considered next, including thickness, coarseness, Hamiltonian cycles and induced paths and cycles. Finally, various embedding and packing problems are discussed, including the determination of the cubical dimension of a given cubical graph.

Eccentricity and centrality in networks

Abstract The classic concept of centrality discovered by Camille Jordan in the 19th century is introduced as a model for social network analysis. It is generalized to include the path center of a graph and illustrated with an application to two island networks in Oceania. It is shown to be a necessary addition to the concepts of degree, closeness and betweenness centrality as distinguished by Freeman.

On the corona of two graphs

Our object in this note is to construct a new and simple operation on two graphs Gt and G2, called their corona, with the property that the group of the new graph is in general isomorphic with the wreath product of the groups of Ga and of G2. Consider two permutat ion groups A and B of order m and n respectively acting on objects sets X = { x 1, x2 ..... Xd} and Y={Yl,Yz,. . . ,Y~}. By their composition or wreath product A [B] we will mean that permutat ion group of order mn a acting on X × Y in which each permutat ion c~A and each sequence (/3~,/~2 . . . . . /~d) of permutations in B induce the permutat ion y =(~;/~1, f12 . . . . . /3a) such that Y(xi, y j) =(c~xi,/~iyj). We write A = B to mean that two permutat ion groups A and B are not only isomorphic but also permutat ionally equivalent. More specifically let h : A ~ B be an isomorphism. To define A = B, we also require another 1 1 map f : X~--~Y between the objects such that for all x in X and c~ in A, f (~x )=h(~) f (x ) . Let graphs 2) G 1 and G 2 have point sets 1/1 and V 2. Then GI[G2], their lexicographic product, has V1 x V2 as its set of points, with two of its points u = (u~, uz) and v =(v l , vz) adjacent whenever ul is adjacent to v~ in Gi, or Ux = v l and u2 is adjacent to v2 in G2. Let A~ and Az be the groups of graphs G~ and G2; then necessary and sufficient condit ions for A ~ CA/] to be permutationally equivalent with the group of GI [G2] were found by Sabidussi [2]. In order to state this result, we recall from [1] that F(G) denotes the group of graph G and (7 the complement of G, that the neighborhood N(v) of a point v of G is the set of points adjacent with v, and the closed neighborhood of v is N(v) u {v}.

A formal system for information retrieval from files

A generalized file structure is provided by which the concepts of keyword, index, record, file, directory, file structure, directory decoding, and record retrieval are defined and from which some of the frequently used file structures such as inverted files, index-sequential files, and multilist files are derived. Two algorithms which retrieve records from the generalized file structure are presented.

The Cohesiveness of Blocks In Social Networks: Node Connectivity and Conditional Density

This study shows several ways that formal graph theoretic statements map patterns of network ties into substantive hypotheses about social cohesion. If network cohesion is enhanced by multiple connections between members of a group, for example, then the higher the global minimum of the number of independent paths that connect every pair of nodes in the network, the higher the social cohesion. The cohesiveness of a group is also measured by the extent to which it is not disconnected by removal of 1, 2, 3, . . . , k actors. Mengers Theorem proves that these two measures are equivalent. Within this graph theoretic framework, we evaluate various concepts of cohesion and establish the validity of a pair of related measures:.

A Survey of combinatorial theory

Abstract : The textbook contains survey articles and research papers covering almost all areas of combinatorial mathematics, in particular graph theory, finite geometries, block designs, factorial designs, coding theory, number theory, combinatorial geometries, search theory, communication and computer science problems and combinatorial problems in statistical inference.

On the Geodetic Number of a Graph

For two vertices u and v of a graph G, the set I(u, v) consists of all vertices lying on some u − v geodesic in G. If S is a set of vertices of G, then I(S) is the union of all sets I(u, v) for u, v ∈ S. The geodetic number g(G) is the minimum cardinality among the subsets S of V(G) with I(S) = V(G). It is shown that if G is a graph of order n and diameter d then g(G) ≤ n − d + 1 and this bound is sharp. For positive integers r, d, and k ≥ 2 with r ≤ d ≤ 2r, there exists a connected graph G of radius r, diameter d, and g(G) = k. Also, for integers n, d, and k with 2 ≤ d < n, 2 ≤ k < n, and n − d − k + 1 ≥ 0, there exists a graph G of order n, diameter d, and g(G) = k. It is shown, for every nontrivial connected graph G, that g(G) ≤ g(G × K2). A sufficient condition for the equality of g(G) and g(G × K2) is presented. A graph F is a minimum geodetic subgraph if there exists a graph G containing F as an induced subgraph such that V(F) is a minimum geodetic set for G. Minimum geodetic subgraphs are characterized.

COVERING AND PACKING IN GRAPHS, I.

T h e r e i s cons iderable i n t e r e s t i n two r e l a t e d types of graph theore t ic problems. m i n i m u m number of g r a p h s with a c e r t a i n p r o p e r t y whose union is a given graph , The cor responding packing p r o b l e m , on the o ther hand , s e e k s the m a x i m u m number of subgraphs of a given g r a p h which have the opposite proper ty . subgraphs a r e r e g a r d e d as s e t s of l ines . s tudy of t h e s e two types A cover ing problem involves finding the

On double and multiple interval graphs

In this paper we discuss a generalization of the familiar concept of an interval graph that arises naturally in scheduling and allocation problems. We define the interval number of a graph G to be the smallest positive integer t for which there exists a function f which assigns to each vertex u of G a subset f(u) of the real line so that f(u) is the union of t closed intervals of the real line, and distinct vertices u and v in G are adjacent if and only if f(u) and f(v)meet. We show that (1) the interval number of a tree is at most two, and (2) the complete bipartite graph Km, n has interval number ⌈(mn + 1)/(m + n)⌉.

Generalized Ramsey theory for graphs

The classical Ramsey numbers [7] involve the occurrence of monochromatic complete subgraphs in line-colored complete graphs. By removing the completeness requirements and admitting arbitrary forbidden subgraphs within any given graph, the situation is richly and nontrivially generalized. The Ramsey number r(m, n) as traditionally studied in graph theory [5, p. 15] may be defined as the minimum number p such that every graph with p points which does not contain the complete graph Km must have n independent points. Alternatively, it is the smallest p for which every coloring of the lines of Kp with two colors, green and red, contains either a green Km or a red Kns Thus the diagonal Ramsey numbers r(n, n) can be described in terms of 2-coloring the lines of Kp and regarding Kn as a forbidden monochromatic subgraph without regard to color. This viewpoint suggests the more general situation in which an arbitrary graph G has a c-coloring of its lines and the number of monochromatic occurrences of a forbidden subgraph F (or of a forbidden family of graphs) is calculated. A host of problem areas within graph theory can be subsumed under such a formulation. These include the line-chromatic number, in which the 3-point path is forbidden. The arboricity of G involves forbidding all cycles. The thickness of a graph forbids the Kuratowski graphs. Complete bipartite graphs can be taken for both G and F, and so can cubes Qn and Qm. There has long been a sentiment in graph theory that there is an intimate relationship between extremal graph theory and Ramsey numbers. It does not appear possible to derive either Turâns theorem or Ramseys theorem from the other. However, we show that extremal bipartite graph theory does in fact imply the bipartite form of Ramseys theorem. Let 3F be a family of graphs, G a given graph, and c a positive integer. We denote by R(G, &, c) the greatest integer n with the property that, in every coloration of the lines of G with c colors, there are at least n monochromatic occurrences of a member of